Communication Systems and Methods of Communicating Utilizing Cooperation Facilitators

ABSTRACT

Systems and methods for improved data transmission utilizing a communication facilitator are described in accordance with embodiments of the invention. One embodiment includes a plurality of nodes, that each comprise: a transmitter; a receiver; and an encoder that encodes message data for transmission using a plurality of codewords; a cooperation facilitator node comprising: a transmitter; and a receiver; wherein the nodes are configured to transmit data parameters to the cooperation facilitator; wherein the cooperation facilitator is configured to generate cooperation parameters based upon the data parameters received from the nodes; wherein the cooperation facilitator is configured to transmit cooperation parameters to the nodes; and wherein the encoder in each of the nodes selects a codeword from the plurality of codewords based at least in part upon the cooperation parameters received from the communication facilitator and transmit the selected codeword via the multiple access channel.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present invention claims priority to U.S. Provisional PatentApplication Ser. No. 62/174,026 entitled “Cooperation Facilitator” toNoorzad et al., filed Jun. 11, 2015, and U.S. Provisional PatentApplication Ser. No. 62/281,636 entitled “Cooperation Facilitator” toNoorzad et al., filed Jan. 21, 2016. The disclosure of U.S. ProvisionalPatent Application Ser. No. 62/174,026 and U.S. Provisional PatentApplication Ser. No. 62/281,636 are herein incorporated by reference intheir entirety.

FIELD OF THE INVENTION

The present invention generally relates to data transmission and morespecifically relates to a cooperative coding scheme to achieve improveddata transmission.

BACKGROUND

Data transmission is the process of sending information in a computingenvironment. The speed at which data can be transmitted is its rate.Data rates are generally measured in megabits or megabytes per second.As consumer use of larger amounts of data increases (for example anincrease in the quality of on demand video streaming and/or downloadinglarger and larger data files), a demand for increased data ratessimilarly increases.

Data transmission can be wired or wireless. Wired communicationprotocols transmit data over a physical wire or cable and can include(but are not limited to) telephone networks, cable television networks(which can transmit cable television and/or Internet services), and/orfiberoptic communication networks. Wireless communication protocols onthe other hand transmit data without a physical wire and can include(but are not limited to) radio, satellite television, cellular telephonetechnologies (such as Long-Term Evolution (LTE)). Wi-Fi, and Bluetooth.Many networks incorporate both wireline and wireless communication. Forexample, the internet can be accessed by either wireline or wirelessconnections.

SUMMARY OF THE INVENTION

An important challenge in communication systems engineering is thegrowing demand for network resources. One way to increase networkperformance in accordance with various embodiments of the invention isto enable cooperation in the network, that is, to allow some networknodes to help others achieve improved performance, such as highertransmission rates or improved reliability. Any network node thatenables other nodes to cooperate can be referred to as a cooperationfacilitator (CF). One metric that can be used to evaluate the benefitsof using a CF is a metric referred to herein as sum-capacity. Thesum-capacity of a network is the maximum amount of information that ispossible to transmit over that network. Cooperation gain can be definedas the difference between the sum-capacity of a network with cooperationand the sum-capacity of the same network without cooperation. In manyinstances, the cooperation gain of coding strategies in accordance withvarious embodiments of the invention grows faster than any linearfunction, when viewed as a function of the total number of bits the CFshares with the transmitters. This means that a small increase in thenumber of bits shared with the transmitters results in a largecooperation gain. It is important to note, that the benefits obtainedusing a CF are not limited to wireline/wireless communications, but mayalso include a variety of other areas where information theory isfrequently used, such as data storage. It is likewise important to notethat increasing sum-capacity is not the only potential benefit ofcooperation: a variety of other benefits are possible including improvedreliability and increased individual rates.

Systems and methods for improved data transmission utilizing acommunication facilitator are described in accordance with embodimentsof the invention. One embodiment includes a plurality of nodes, thateach comprise: a transmitter; a receiver; and an encoder that encodesmessage data for transmission using a plurality of codewords; acooperation facilitator node comprising: a transmitter, and a receiver;wherein the plurality of nodes are configured to transmit dataparameters to the cooperation facilitator; wherein the cooperationfacilitator is configured to generate cooperation parameters based uponthe data parameters received from the plurality of nodes; wherein thecooperation facilitator is configured to transmit cooperation parametersto the plurality of nodes; and wherein the encoder in each of theplurality of nodes selects a codeword from the plurality of codewordsbased at least in part upon the cooperation parameters received from thecommunication facilitator and transmit the selected codeword via themultiple access channel.

In a further embodiment, a sum-capacity of the communication systemachieved using codewords selected at least in part based upon thecooperation parameters received from the cooperation facilitator isgreater than the sum-capacity of the communication system achieved wheneach of the plurality of encoders encodes data without communicatingwith a cooperation facilitator.

In another embodiment, a reliability of the communication systemachieved using codewords selected at least in part based upon thecooperation parameters received from the cooperation facilitator isgreater than the reliability of the communication system achieved witheach of the plurality of encoders encodes data without communicatingwith a cooperation facilitator.

In a still further embodiment, the cooperation parameters includeconferencing parameters.

In still another embodiment, the cooperation parameters includecoordinating parameters.

In a yet further embodiment, the transmitter in each of the plurality ofnodes transmits data via a multiple access channel.

In yet another embodiment, the multiple access channel is a sharedwireless channel.

In a further embodiment again, the multiple access channel is a Gaussianmultiple access channel.

In another embodiment again, the plurality of nodes is two nodes.

In a further additional embodiment, the plurality of nodes is at leastthree nodes.

In another additional embodiment, the transmitter in each of theplurality of nodes transmits to a plurality of receivers.

In a still yet further embodiment, the cooperation facilitator generatesmultiple rounds of cooperation parameters prior to codewordtransmission.

In still yet another embodiment, cooperation parameters are transmittedto the plurality of nodes by the coordination facilitator via a separatechannel to a channel on which one or more of the plurality of nodestransmit codewords selected at least in part based upon the cooperationparameters received from the cooperation facilitator.

In a still further embodiment again, a cooperation facilitator,comprising: a transmitter; a receiver; and a cooperation facilitatorcontroller; wherein the cooperation facilitator controller is configuredto receive data parameters from a plurality of nodes; wherein thecooperation facilitator is configured to generate cooperation parametersbased upon the data parameters received from the plurality of nodes; andwherein the cooperation facilitator is configured to transmitcooperation parameters to the plurality of nodes that enable encoders ineach of the plurality of nodes to select a codeword from a plurality ofcodewords for transmission.

In still another embodiment again, a sum-capacity of a portion of acommunication network including the cooperation facilitator achieved byencoders in each of the plurality of nodes using codewords selected atleast in part based upon the cooperation parameters received from thecooperation facilitator is greater than the sum-capacity of the portionof a communication network achieved when each of the plurality ofencoders encodes data without communicating with a cooperationfacilitator.

Another further embodiment of the method of the invention includes: areliability of a portion of a communication network including thecooperation facilitator achieved by encoders in each of the plurality ofnodes using codewords selected at least in part based upon thecooperation parameters received from the cooperation facilitator isgreater than the reliability of the portion of a communication networkachieved with each of the plurality of encoders encodes data withoutcommunicating with a cooperation facilitator.

Still another further embodiment of the method of the inventionincludes: the cooperation parameters include conferencing parameters.

In a further embodiment again, the cooperation parameters includecoordinating parameters.

In another embodiment again, a transmitter in each of the plurality ofnodes transmits data via a multiple access channel.

In a further additional embodiment, the multiple access channel is ashared wireless channel.

In another additional embodiment, the multiple access channel is aGaussian multiple access channel.

In a still yet further embodiment, the plurality of nodes is two nodes.

In still yet another embodiment, the plurality of nodes is at leastthree nodes.

In a still further embodiment again, the cooperation facilitatorgenerates multiple rounds of cooperation parameters prior to codewordtransmission.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating a system incorporating a cooperationfacilitator for at least two other transmitters sharing a multipleaccess channel in accordance with an embodiment of the invention.

FIG. 2 is a diagram conceptually illustrating cooperation among networknodes in accordance with an embodiment of the invention.

FIG. 3 is a diagram illustrating a cooperation facilitator systemutilizing a multiple access channel in accordance with an embodiment ofthe invention.

FIG. 4 is a block diagram illustrating a cooperation facilitatorcontroller for data transmission applications in accordance with anembodiment of the invention.

FIG. 5 is a flow chart illustrating a coordinated data transmissionprocess in accordance with an embodiment of the invention.

FIG. 6 is a flow chart illustrating a cooperation facilitator process inaccordance with an embodiment of the invention.

FIG. 7 is a flow chart illustrating an encoding process in accordancewith an embodiment of the invention.

FIG. 8 is a plot illustrating maximum sum-rate gain achieved for aGaussian multiple access channel in accordance with an embodiment of theinvention.

FIG. 9 is a diagram illustrating a k-user multiple access channelutilizing a cooperation facilitator in accordance with an embodiment ofthe invention.

FIG. 10 is a plot illustrating a comparison of the achievable sum-ratefor different networks utilizing a cooperation facilitator in accordancewith an embodiment of the invention.

FIG. 11 is a diagram illustrating a k-user multiple access channel withconferencing in accordance with an embodiment of the invention.

FIG. 12A is a diagram illustrating a conferencing structure for a 3-userGaussian multiple access channel in accordance with an embodiment of theinvention.

FIG. 12B is a diagram illustrating an alternative conferencing structurefor a multiple access channel in accordance with an embodiment of theinvention.

FIG. 13 is a diagram illustrating a memoryless multiple access channeland a cooperation facilitator in accordance with an embodiment of theinvention.

DETAILED DESCRIPTION

As an example, we now turn to the drawings, systems and methods ofimproving data transmission over a multiple access channel (MAC) byutilizing communication between encoders through a cooperationfacilitator (CF) in accordance with various embodiments of the inventionare illustrated. Data networks such as (but not limited to) a cellularnetwork that rely upon MACs have limits on the amount of data that canbe sent or received by the multiple transmitters accessing the channel.In many embodiments, communication between the transmitters via acooperation facilitator device that coordinates or otherwise interactswith the encoding of transmitted data by encoders within thetransmitters can achieve significant improvements in the performance ofthe network. In certain embodiments, coordination by a cooperationfacilitator can increase the rate of a MAC. In several embodiments,transmission reliability can also be increased (in addition totransmission rate). While communication between a cooperationfacilitator and multiple transmitters or other devices within a networkcan utilize system resources, in many embodiments an overall improvementin the performance of the network that incorporates the MAC can beachieved. The benefits of using a CF to coordinate encoding by thetransmitters are primarily achieved, because each cooperatingtransmitter only needs to transmit a message including information aboutitself. The CF can transmit a message that is based upon the messagesreceived from the cooperating transmitters to facilitate cooperationbetween the transmitters to coordinate the encoders in the transmitters.The transmitters send an encoding of their message to the CF, whichcalculates a combined encoding to produce messages that coordinate thecooperation to the transmitters. As is discussed in detail below, whenthe cooperation is coordinated effectively the benefits of thecooperation can result in an increase in evaluation metrics. Forexample, for capacity, the benefits of cooperation for the transmittersexceeds the cost of sending data to facilitate cooperation.

While much of the discussion that follows relates to systems in whichtwo encoders communicate with a cooperation facilitator, in many otherembodiments more than two encoders communicate with a cooperationfacilitator. In addition, communication can be between multipletransmitters and a single receiver or between multiple transmitters andmultiple receivers or between transmitters and/or receivers and devicesthat are both transmitters and receivers in accordance with variousembodiments of the invention. In several embodiments, encoders passportions of the messages they are encoding to the CF. The CF can passportions of these messages unaltered to other encoders in the system.Additionally, the CF can “coordinate” transmissions, i.e. enable theencoders to create dependence among independently generated codewords.In several embodiments, encoders can utilize information from the CF aswell as portions of the input message which did not go through the CF togenerate the encoded message. Communication systems and methods ofcommunicating utilizing cooperation facilitators in accordance withvarious embodiments of the invention are discussed further below.

Cooperation Facilitator Systems

An example communication system incorporating a device that acts as acooperation facilitator for at least two other transmitters sharing aMAC in accordance with an embodiment of the invention is illustrated inFIG. 1. Cellular devices 102, 104, and 106 connect to cellular tower 108via a wireless connection 110. These cellular devices can include (butare not limited to) cellular phones, tablets modems, and/or basestations. Cellular tower 108 connects to network 112 via wiredconnection 114. Network 112 can be (but is not limited to) a largercellular telephone network and/or the Internet with potential wiredand/or wireless connections to devices 102, 104 and/or 106. As canreadily be appreciated, a larger cellular telephone network wouldinclude many more cellular towers and involve a very large number ofMACs and other communication components shared by the cellularsubscriber devices that utilize the network.

At any given time, there is a maximum amount of data that can betransmitted to and from a specific cellular tower 108 by cellulardevices 104 and/or 106. This can be referred to as the throughput of thenetwork. For example, when a large number of cellular devices try toreceive data from the cellular tower at the same time, download speedswill be slower compared to a single device trying to receive data fromthe cellular tower. In various embodiments of the invention, cellulardevice 102 acts as a cooperation facilitator (CF) to coordinate datatransmission by cellular devices 104, 106 and 108. In many embodimentsof the invention, utilizing some of the available bandwidth tocoordinate the transfer of other data can greatly increase thethroughput and/or reliability of the transfer of that other data.Cooperation among network nodes is further described below andconceptually illustrated below in FIG. 2.

In many embodiments of the invention, the CF device can communicate withother cellular devices to coordinate data transfer over (but not limitedto) the cellular network. Bluetooth, and/or WIFI. In several otherembodiments, a CF can be software running on one or more cellulardevices. It should readily be apparent that the use of a cellularnetwork is merely illustrative, and a cooperation facilitator can beutilized in a variety of applications to improve data transmissions.Although a variety of cooperation facilitator systems are describedabove with reference to FIG. 1, any of a variety of networks forimproving data transmission can be utilized as appropriate to therequirements of specific applications in accordance with variousembodiments of the invention. Cooperation among network nodes isdiscussed below.

FIG. 2 conceptually illustrates cooperation among network nodes.Cooperation facilitator node C 206 facilitates communication betweennode A 202, node B 204 and network 208. Node A 202 passes information tonode C 206 via connection 210. This connection (and all connections inthis conceptual illustration) can be wired and/or wireless. Similarly,node B 204 passes information to node C 206 via connection 212. CF nodeC 206 coordinates data transmission and passes information to node A 202via connection 214 and node B 204 via connection 216. Specifics of datatransmission to and from the CF will be described in greater detailbelow and can include (but is not limited to) portions of data beingpasses from node A to node B via node C without further calculations aswell as the results of calculations performed on data by the CF. In acoordinated manner, node A 202 passes information to network 208 viaconnection 218, and node B 204 passes information to network 208 viaconnection 220.

FIG. 3 illustrates a cooperation facilitator system utilizing a multipleaccess channel (MAC) in accordance with several embodiments of theinvention. Encoder one 302 receives message one 304. Similarly, encodertwo 306 receives message two 308. Cooperation Facilitator (CF) 310receives data from encoder one 302 via connection 312 and from encodertwo 304 via connection 314. Connections 312 and 314 (as well as allother connections in FIG. 3) can be either wired and/or wireless. CF 310passes information to encoder one 302 via connection 316 and to encodertwo via connection 318. Information passed to and from CF 310 will bediscussed in greater detail below and can include (but is not limitedto) portions of information being passed from encoder one to encoder twovia the CF without further calculations as well as the results ofcalculations performed on data by the CF.

In several embodiments of the invention, encoders can transmitinformation utilizing a multiple access channel (MAC). In variousembodiments, the MAC can be (but is not limited to) a Gaussian MAC or adiscrete memoryless MAC. Encoder one 302 connects to MAC 320 viaconnection 322. Similarly, encoder two 306 connects to MAC 320 viaconnection 324. In several embodiments, a cellular tower as describedabove with respect to FIG. 1 can transmit information utilizing a MAC.It should readily be apparent that embodiments of the invention are notlimited to cellular telephone networks and other implementations can beutilized as appropriate to the requirements of specific communicationnetworks. Decoder 326 receives information from MAC 320 via connection328. Decoder 326 generates decoded message one 330 and decoded messagetwo 332.

Although a variety of cooperation facilitator systems are describedabove with reference to FIG. 3, any of a variety of networks utilizingcooperation facilitators to improve data transmission can be utilized asappropriate to the requirements of specific applications in accordancewith various embodiments of the invention. Cooperation Facilitatorcontrollers are described below.

Cooperation Facilitator Controllers

Cooperation Facilitator controllers which implement data transmissionapplications in accordance with many embodiments of the invention aredescribed in FIG. 4. In several embodiments, Cooperation Facilitator(CF) controller 400 can coordinate the transmission of data among nodesin a network to improve performance including rate and/or reliability.The CF controller includes at least one processor 402, an I/O interface404, and memory 406. In many embodiments, the memory 406 includessoftware including data transmission application 408, as well as dataparameters 410, conferencing parameters 412, and coordinating parameters412. In several embodiments, data parameters 410 can include part of themessage data from each node (or encoder) that is being encoded by thesystem. Conferencing parameters 412 can include information passedunaltered from one node to another through the CF controller.Coordinating parameters are generated by the CF to enable coordinateddata transmission. Generally they enable encoders within a system tocreate dependence among independently generated codewords. In manyembodiments, the results of rate-distortion theory can be utilized togenerate coordinating parameters. Data parameters, conferencingparameters, and coordinating parameters will be discussed in greaterdetail below. Although a variety of CF controllers are described abovewith reference to FIG. 4, any of a variety of controllers forcooperative data transmission can be utilized as appropriate to therequirements of specific applications in accordance with embodiments ofthe invention. Processes for cooperative data transmission will bediscussed below.

Coordinated Data Transmission Processes

An overview of a coordinated data transmission process 500 that utilizesa cooperation facilitator in accordance with several embodiments of theinvention is illustrated in FIG. 5. Data parameters are passed 502 fromencoders to the cooperation facilitator. Generally, data parameters arepart of the messages being encoded by the system. In various embodimentstwo encoders are utilized. A CF system utilizing two encoders will bedescribed in greater detail below. In several other embodiments, morethan two encoders are utilized. A CF system utilizing more than twoencoders will also be described in greater detail below. The CFdetermines 504 conferencing parameters and coordinating parameters.Conferencing parameters are generally pieces of data passed unalteredbetween encoders through the CF, and will be discussed further below.Coordinating parameters are generated by the CF to enable coordinateddata transmission. Generally they enable encoders within a system tocreate dependence among independently generated codewords. In manyembodiments, the results of rate-distortion theory can be utilized togenerate coordinating parameters. The generation of coordinatingparameters will be discussed in greater detail below. The CF sends 506conferencing parameters and coordinating parameters to encoders.Encoders encode 508 message data utilizing conferencing parameters andcoordinating parameters in addition to portions of the input messagethat do not pass through the CF. Data is transmitted 510 over a multipleaccess channel (MAC) to one or more decoders. The decoder decodes 512the data to reconstruct the message data.

In many embodiments, the use of a CF can improve data transmission. Insome embodiments, for example in noisy environments, the rate of datatransmission can be increased. In many other embodiments, thereliability of data transmission can be increased. Increase in data rateand increase in data reliability will be discussed in further detailbelow. Although a variety of data transmission processes are describedabove with reference to FIG. 5, any of a variety of processes forcoordinated data transmission can be utilized as appropriate to therequirements of specific applications in accordance with embodiments ofthe invention. Cooperation Facilitator processes are described below.

Cooperation Facilitator Processes

A cooperation facilitator process 600 that can be performed by acooperation facilitator node to coordinate data transmission inaccordance with an embodiment of the invention is illustrated in FIG. 6.The CF receives 602 data parameters from encoders. Conferencingparameters are determined 604 from data parameters. In many embodiments,conferencing parameters pieces of data passed unaltered between encodersthrough the CF, and will be discussed further below. Coordinatingparameters are determined 606 from data parameters. In severalembodiments, coordinating parameters are generated by the CF to enablecoordinated data transmission within the system. Generally they enableencoders within a system to create dependence among independentlygenerated codewords, and are discussed in greater detail below.Conferencing parameters and coordinating parameters are sent 608 toencoders. In various embodiments, two encoders are utilized. In manyembodiments, more than two encoders are utilized. Specific details forapplications with two encoders and more than two encoders are discussedin greater detail below. Although a variety of cooperation facilitatorprocesses are described above with reference to FIG. 6, any of a varietyof processes to coordinate cooperation in a data transmission networkcan be utilized as appropriate to the requirements of specificapplications in accordance with embodiments of the invention. Encodingprocesses are described below.

Encoding Processes

An encoding process 700 that utilizes data from a cooperationfacilitator in accordance with various embodiments of the invention isillustrated in FIG. 7. Data parameters are determined 702 from inputdata. Input data generally is the message and/or messages to be encoded.Data parameters, which will be described in greater detail below, areportions of message data passed to the CF and are utilized in thecooperation facilitator process. Data parameters are passed 704 to theCF. Conferencing parameters are received 706 from other encoders whichare passed via the CF. In many embodiments, conferencing parameterspieces of data passed unaltered between encoders through the CF, andwill be discussed further below. Coordinating parameters are received708 from the CF. In several embodiments, coordinating parameters aregenerated by the CF to enable coordinated data transmission within thesystem. Generally they enable encoders within a system to createdependence among independently generated codewords. Conferencingparameters and coordinating parameters will be discussed in greaterdetail below. Input data is encoded 710 using conferencing parametersand coordinating parameters in addition to the remaining portions ofinput data not passed through the CF. In some embodiments, systemsutilize two encoders. In many other embodiments, systems utilize morethan two encoders. Systems with two encoders and more than two encoderswill be described in detail below. Although a variety of encodingprocesses are described above with reference to FIG. 7, a variety ofprocesses to coordinate data transmission can be utilized as appropriateto the requirements of specific applications in accordance withembodiments of the invention.

1. Gaussian Multiple Access Channels

In several embodiments of the invention, in cooperative coding schemesnetwork nodes work together to achieve higher transmission rates. Toobtain a better understanding of cooperation, consider an embodiment ofthe invention in which two transmitters send rate-limited descriptionsof their messages to a “cooperation facilitator”, a node that sends backrate-limited descriptions of the pair to each transmitter. Thisembodiment of the invention includes the conferencing encoders model. Itcan be shown that except for a special class of multiple accesschannels, the gain in sum-capacity resulting from cooperation under thismodel is quite large. Adding a cooperation facilitator to any suchchannel results in a network that does not satisfy the edge removalproperty. That is, removing a connection of capacity C may decrease thesum-capacity of the network by more than the capacity C. An importantspecial case in accordance with many embodiments of the invention is theGaussian multiple access channel, for which the sum-rate cooperationgain will be explicitly characterized below.

To meet the growing demand for higher transmission rates, network nodesshould employ coding schemes that use scarce resources in a moreefficient manner. By working together, network nodes can take advantageof under-utilized network resources to help data transmission in heavilyconstrained regions of the network. Cooperation among nodes emerges as anatural strategy towards this aim.

As an illustrative example, consider two nodes. A and B, transmittingindependent messages over a network N. A third node C that hasbidirectional links to A and B can help A and B work together to achievea higher sum-rate than they would have achieved had they workedseparately.

In various embodiments, an understanding of how the gain in sum-rateresulting from cooperation between A and B relates to the capacities ofthe links from (A,B) to C and back is important. Intuitively, theincrease in sum-rate can be thought of as the benefit of cooperation andthe capacities of the links between (A,B) and C as the cost ofcooperation. See FIG. 2 as described above, which illustratescooperation among network nodes. Node C enables nodes A and B tocooperate and potentially achieve higher rates in the transmission oftheir messages over network N.

To study this embodiment of the invention formally, let A and B be theencoders of a memoryless multiple access channel (MAC). Let C be a“cooperation facilitator” (CF), a node which, prior to the transmissionof the messages over the network, receives a rate-limited description ofeach encoder's message and sends a rate-limited output to each encoder.See FIG. 3 as described above which illustrates a network for the MACwith a CF.

In one-step cooperation, each encoder sends a function of its message tothe CF and the CF transmits, to each encoder, a value that is a functionof both of its inputs. Similarly, k-step cooperation (for a fixedpositive integer k) can be defined between the CF and the encoders wherethe information transmission between the CF and each encoder continuesfor k steps, with the constraint that the information that the CF oreach encoder transmits in each step only depends on the information thatit previously received. Only one-step cooperation is used for simplicityin the achievability result.

The CF of several embodiments of the invention extends the cooperationmodel to allow for rate-limited inputs. While the CF in earlierapproaches has full knowledge of both messages and transmits arate-limited output to both encoders, the more general CF of manyembodiments of the invention only has partial knowledge of eachencoder's message. In addition, the CF can be allowed to send adifferent output to each encoder.

There exists a discrete memoryless MAC where encoder cooperation througha CF results in a large gain (with respect to the capacities of theoutput edges of the CF). This implies the existence of a networkconsisting of a MAC with a CF that does not satisfy the “edge removalproperty”. A network satisfies the edge removal property if removing anedge from that network does not reduce the achievable rate of any of thesource messages by more than the capacity of that edge. A questionexists as to whether such a result is true for more natural channels,e.g., the Gaussian MAC. The answer turns out to be positive, and exceptfor a special class of MACs, adding a CF results in a large sum-capacitygain.

An achievability scheme in accordance with many embodiments of thepresent invention combines three coding schemes via rate splitting.First, each encoder sends part of its message to the CF. The CF passeson part of what it receives from each encoder to the other encoderwithout any further operations. In this way the CF enables“conferencing” between the encoders, which is a cooperation strategy.

The CF uses the remaining part of what it receives to help the encoders“coordinate” their transmissions; that is, it enables the encoders tocreate dependence among independently generated codewords. For thiscoordination strategy, results from rate-distortion theory can be reliedupon to obtain an inner bound for the capacity region of the broadcastchannel.

Finally, for the remaining part of the messages, which do not go throughthe CF, the encoders can use a classical coding scheme. The achievablescheme is more formally introduced below and its performance is studiedfurther below. An inner bound for the Gaussian MAC is provided. Thesum-rate gain of the inner bound is compared with the sum-rate gain ofother schemes. No other scheme alone performs as well as combinations inaccordance with many embodiments of the inventions.

1.1 Cooperation Models

Let (χ₁×χ₂,P(y|x₁,x₂),ψ) denote a memoryless MAC. Suppose W₁ and W₂ arethe messages that encoders 1 and 2 transmit, respectively. For everypositive integer k, define [k]={1, . . . , k}. Assume that W₁ and W₂ areindependent and uniformly distributed over the sets [M₁] and [M₂],respectively.

For i=1,2, represent encoder i by the mappings

φ_(i) :[M _(i)]→[2 ^(nC) ^(i) ^(in) ]

ƒ_(i) :[M _(i)]×[2^(nC) ^(i) ^(out) ]→χ_(i) ^(n)

that describe the transmissions to the CF and channel, respectively.Represent the CF by the mappings

ψ_(i):[2 ^(nC) ¹ ^(in) ]×[2 ^(nC) ² ^(in) ]→[2 ^(nC) ¹ ^(out) ],

where ψ_(i) denotes the output of the CF to encoder i for i=1,2. Underthis definition, when (W₁, W₂)=(w₁, w₂), the CF receives φ₁ (w₁) andφ₂(w₂) from encoders 1 and 2, respectively. The CF then sends ψ₁(φ₁(w₁), φ₂(w₂)) to encoder 1 and ψ₂ (φ₁(w₁), φ₂(w₂)) to encoder 2.

Represent the decoder by the mapping

g:ψ ^(n) →[M ₁ ]×[M ₂].

Then the probability of error is given by

P _(e) ^((n)) =P{g(Y ^(n))≠(W ₁ ,W ₂)}.

Define C^(in)=(C₁ ^(in),C₂ ^(in)) and C^(out)=(C₁ ^(out),C₂ ^(out)).Call the mappings (φ₁,φ₂,ψ₁,ψ₂,ƒ₁,ƒ₂,g) an (n,M₁,M₂) code for the MACwith a (C^(in),C^(out))−CF. For nonnegative real numbers R₁ and R₂, saythat the rate pair (R₁,R₂) is achievable if for every ∈>0 andsufficiently large n, there exists an (n, M₁,M₂) code such that P_(e)^((n))≦∈ and

${{\frac{1}{n}\log \; M_{i}} > {R_{i} - ɛ}},$

for i=1,2. We define the capacity region as the closure of the set ofall achievable rate pairs (R₁,R₂) and denote it by

(C^(in),C^(out))

Using the capacity region of the MAC with conferencing encoders, obtaininner and outer bounds for the capacity region of a MAC with a CF. Let

_(conf)(C₁₂,C₂₁) denote the capacity region of a MAC with a (C₁₂,C₂₁)conference. Since the conferencing capacity region can be achieved witha single step of conferencing, it follows that

_(conf)(min{C ₁ ^(in) ,C ₂ ^(out)},min{C ₂ ^(in) ,C ₁ ^(out)})

is an inner bound for

(C^(in),C^(out)). In addition, since each encoder could calculate the CFoutput if it only knew what the CF received from the other encoder,

_(conf)(C₁ ^(in),C₂ ^(in)) is an outer bound for

(C^(in),C^(out)). Henceforth refer to these inner and outer bounds asthe conferencing bounds. Note that when C₂ ^(out)≧C₁ ^(in) and C₁^(out)≧C₂ ^(in), the conferencing inner and outer bounds agree, giving

(C ^(in) ,C ^(out))=

_(conf)(C ₁ ^(in) ,C ₂ ^(in)).

Next discuss the main result of this section. For any memoryless MAC(χ₁×χ₂,P(y|x₁,x₂),ψ) with a (C^(in),C^(out))−CF, define the sum-capacityas

$C_{sum} = {\max\limits_{{({C_{in},C_{out}})}}{\left( {R_{1} + R_{2}} \right).}}$

For a fixed C_(in) with min{C₁ ^(in),C₂ ^(in)}>0, define the“sum-capacity gain” G:

_(≧0)→

_(≧0) as

G(C _(out))=C _(sum)(C ^(in) ,C ^(out))−C _(sum)(C ^(in),0),

where C_(out)=(C_(out),C_(out)) and 0=(0,0). Note that when C_(out)=0,no cooperation is possible, thus

${C_{sum}\left( {C^{in},0} \right)} = {\max\limits_{{P{(x_{1})}}{P{(x_{2})}}}{{I\left( {X_{1},{X_{2};Y}} \right)}.}}$

It can be proven that for any MAC where using dependent codewords(instead of independent ones) results in an increase in sum-capacity,the effect of cooperation through a CF can be quite large. Inparticular, it shows that the network consisting of any such MAC and aCF does not satisfy the edge removal property.

Theorem 1 (Sum-Capacity).

For any discrete memoryless MAC (χ₁×χ₂,P(y|x₁,x₂),ψ) that satisfies

${{\max\limits_{P{({x_{1},x_{2}})}}{I\left( {X_{1},{X_{2};Y}} \right)}} > {\max\limits_{{P{(x_{1})}}{P{(x_{2})}}}{I\left( {X_{1},{X_{2};Y}} \right)}}},$

we have G′(0)=∞. For the Gaussian MAC, a stronger result holds: For somepositive constant α and sufficiently small C_(out),

G(C _(out))≧α√{square root over (C _(out))},

The proof of Theorem 1 can be found in Parham Noorzad, Michelle Effros,and Michael Langberg, On the Cost and Benefit of Cooperation (ExtendedVersion), arxiv.org/abs/1504.04432, 17 Apr. 2015, which is herebyincorporated by reference in its entirety, and is based on anachievability result for the MAC with a CF, which is next described.Define

(C ^(in) ,C ^(out))

as the set of all rate pairs (R₁,R₂) that for (i,j)∈{(1, 2),(2, 1)}satisfy

R _(i) <I(X _(i) ;Y|U,V ₁ ,V ₂ ,X _(j))+C _(i) ^(in)

R _(i) <I(X _(i) ;Y|U,V _(j) ,X _(i))+C _(i0)

R ₁ +R ₂ <I(X ₁ ,X ₂ ;Y|U,V ₁ ,V ₂)+C ₁ ^(in) +C ₂ ^(in)

R ₁ +R ₂ <I(X ₁ ,X ₂ ;Y|U,V _(i))+C _(i) ^(in) +C _(j0)

R ₁ +R ₂ <I(X ₁ ,X ₂ ;Y|U)+C ₁₀ +C ₂₀

R ₁ +R ₂ <I(X ₁ ,X ₂ ;Y),

for nonnegative constants C₁₀ and C₂₀, and distributions P(u, v₁,v₂)P(x₁|u, v₁)P(x₂|u, v₂) that satisfy

C _(i0)≦min{C _(i) ^(in) ,C _(j) ^(out)}

I(V ₁ ;V ₂ |U)≦(C ₁ ^(out) −C ₂₀)+(C ₂ ^(out) −C ₁₀).  (1)

In the above definition, the pair (U, V_(i)) represents the informationencoder i receives from the CF. In addition, the pair (C₁₀,C₂₀)indicates the amount of rate being used on the CF links to enable theconferencing strategy. The remaining part of rate on the CF links isused to create dependence between V₁ and V₂.

Theorem 2 (Achievability).

For any menmoryless MAC (χ₁×χ₂,P(y|x₁,x₂),ψ) with a (C^(in),C^(out))−CF,the rate region

(C^(in),C^(out)) is achievable.

A nontrivial special case is the case where the CF has completeknowledge of both source messages, that is, C₁ ^(in)=C₂ ^(in)=∞. In thiscase, it is not hard to see in Parham Noorzad, Michelle Effros, andMichael Langberg, On the Cost and Benefit of Cooperation (ExtendedVersion), arxiv.org/abs/1504.04432, 17 Apr. 2015, which is herebyincorporated by reference in its entirety, that

(C^(in),C^(out)) simplifies to the set of all nonnegative rate pairs(R₁,R₂) that satisfy

R ₁ <I(X ₁ :Y|U,X ₂)+C ₁₀

R ₂ <I(X ₂ :Y|U,X ₁)+C ₂₀

R ₁ +R ₂ <I(X ₁ ,X ₂ ;Y|U)+C ₁₀ +C ₂₀

R ₁ +R ₂ <I(X ₁ ,X ₂ ;Y),

for nonnegative constants C₁₀≦C₂ ^(out) and C₂₀≦C₁ ^(out), anddistributions P(u,x₁,x₂) with

I(X ₁ ;X ₂ |U)≦(C ₁ ^(out) −C ₂₀)+(C ₂ ^(out) −C ₁₀).

Note that in this case, increasing the number of cooperation steps doesnot change the family of functions the CF can compute. Thus as with thecase where C₁ ^(in)≦C₂ ^(out) and C₂ ^(in)≦C₁ ^(out), using more thanone step for cooperation does not enlarge the capacity region.

The rate region,

(C^(in),C^(out)), in addition to being achievable, is also convex. Toprove this, we show a slightly stronger result. For every λ∈(0,1),(C_(a) ^(in),C_(a) ^(out)), and (C_(b) ^(in),C_(b) ^(out)), define

_(λ)=

(λC _(a) ^(in)+(1−λ)C _(b) ^(in) ,C _(a) ^(out)+(1−λ)C _(b) ^(out)).

Also define

_(a)=

(C_(a) ^(in),C_(a) ^(out)) and

_(b)=

(C_(b) ^(in),C_(b) ^(out)). We then have the following result.

Theorem 3 (Convexity). For any λ∈(0,1),

_(λ) ⊃λ

_(a)+(1−λ)

_(b).

The addition in Theorem 3 is the Minkowski sum, defined for any twosubsets A and B of

as

A+B={(a ₁ +b ₁ ,a ₂ +b ₂)(a ₁ ,a ₂)∈A,(b ₁ ,b ₂)∈B}.

Set C_(a) ^(in)=C_(b) ^(in) and C_(a) ^(out)=C_(b) ^(out) in Theorem 3to get

⊃λ

+(1−λ)

, which is equivalent to the convexity of

. Using a time-sharing argument, see that the capacity region

(C^(in),C^(out)) also satisfies the property stated in Theorem 3.Theorem 3 is proved in Parham Noorzad, Michelle Effros, and MichaelLangberg, On the Cost and Benefit of Cooperation (Extended Version),arxiv.org/abs/1504.04432, 17 Apr. 2015, which is hereby incorporated byreference in its entirety.

1.2 The Achievability Scheme

In this section, a formal description of a coding scheme that can beutilized in many embodiments of the invention is given. First, picknonnegative constants C₁₀ and C₂₀ such that Equation (1) holds for{i,j}={1,2}. In achievability scheme, the first nC_(i0) bits of W_(i)are sent directly from encoder i to encoder j through the CF without anymodification. Thus require C_(i0) to satisfy inequality (1).

Next, choose C_(1d) and C_(2d) such that

C _(1d) ≦C ₁ ^(out) −C ₂₀

C _(2d) ≦C ₂ ^(out) −C ₁₀.  (2)

The values of C_(1d) and C_(2d) specify the amount of rate used on eachof the output links for the coordination strategy. Finally, choose aninput distribution P(u,v₁,v₂)P(x₁|u,v₁)P(x₂|u,v₂) so that P(u,v₁,v₂)satisfies

ζ:=C _(1d) +C _(2d) −I(V ₁ ;V ₂ |U)>0.  (3)

Fix ∈>0. Let A_(∈) ^((n)) be the weakly typical set with respect to thedistribution

P(u,v ₁ ,v ₂)P(x ₁ |u,v ₁)P(x ₂ |u,v ₂)P(y|x ₁ ,x ₂).

By Cramér's large deviation theorem, there exists a nondecreasingfunction Θ:

→

such that

$\begin{matrix}{{P\left\{ \left( A_{ɛ}^{(n)} \right)^{c} \right\}} \leq {2^{{- n}\; {\Theta {(ɛ)}}}.}} & (4)\end{matrix}$

Fix δ>0 and let A_(δ) ^((n)) denote the weakly typical set with respectto P(u,v₁,v₂). Make use of the typical sets A_(δ) ^((n)) and A_(∈)^((n)) in the encoding and decoding processes, respectively.

Next the codebook generation is described. For i=1,2, let M_(i)=└2 ^(nR)^(i) ┘ and define R_(i0)=min{R_(i),C_(i0)}, R_(id)=min{R_(i),C_(i)^(in)}−R_(i0), and R_(ii)=(R_(i)−C_(i) ^(in))⁺, where for any realnumber x,x⁺=max{x,0}. Note that for i=1,2, R_(i)=R_(i0)+R_(id)+R_(ii),thus each of the messages can be split into three parts as

W _(i)=(W _(i0) ,W _(id) ,W _(ii))∈[2 ^(nR) ^(i0) ]×[2 ^(nR) ^(id) ]×[2^(nR) ^(ii) ].

Here W₁₀ and W₂₀ are used for conferencing. W_(1d) and W_(2d) are usedfor coordination, and W₁₁ and W₂₂ are transmitted over the channelindependently.

Next, for every (w₁₀,w₂₀)∈[2^(n(R) ¹⁰ ^(+R) ²⁰ ⁾], generateU^(n)(w₁₀,w₂₀) i.i.d. with the distribution

${P\left\{ {{U^{n}\left( {w_{10},w_{20}} \right)} = u^{n}} \right\}} = {\prod\limits_{t = 1}^{n}{{P\left( u_{t} \right)}.}}$

Let E(u^(n)) be the event {U^(n)(w₁₀,w₂₀)=u^(n)}. Given E(u^(n)), forevery (w_(id),z_(i))∈[2^(nR) ^(id) ]×[2^(nC) ^(id) ], generate V_(i)^(n)(w_(id),z_(i)|u^(n)) according to

$\begin{matrix}{{{P\left\{ {{V_{i}^{n}\left( {w_{id},{z_{i}u^{n}}} \right)} = {v_{i}^{n}{E\left( u^{n} \right)}}} \right\}} = {\prod\limits_{t = 1}^{n}{P\left( {v_{it}u_{t}} \right)}}},} & \;\end{matrix}$

for i=1,2, where P(v₁|u) and P(v₂|u) are marginals of P(v₁,v₂|u).

Fix (w₁₀,w₂₀,w_(1d),w_(2d)) and functions

v _(i):[2 ^(nC) ^(id) ]→

;

for i=1,2. Let E(u^(n),v₁,v₂) denote the event whereU^(n)(w₁₀,w₂₀)=u^(n) and V₁ ^(n)(w_(1d),•|u^(n))=v₁(•), and V₂^(n)(w_(2d),•|u^(n))=v₂(•). In addition, for any u^(n),v₁, and v₂,define the set

(u ^(n) ,v ₁ ,v ₂):={(z ₁ ,z ₂):(u ^(n) ,v ₁(z ₁),v ₂(z ₂))∈A _(δ)^((n))}.

Given E(u^(n),v₁,v₂), if

(u^(n),v₁,v₂) is nonempty, define

(Z ₁(u ^(n) ,v ₁ ,v ₂),Z ₂(u ^(n) ,v ₁ ,v ₂))

as a random pair that is uniformly distributed on

(u^(n),v₁,v₂). Otherwise, set Z_(i)(u^(n),v₁,v₂)=1 for i=1,2.

Next, fix (w₁₀, w₂₀,w_(1d), w_(2d)) and let E(u^(n),v₁ ^(n),v₂ ^(n))denote the event where U^(n)(w₁₀,w₂₀)=u^(n), V₁ ^(n)(w_(1d),Z₁|u^(n))=v₁^(n) and V₂ ^(n)(w_(2d),Z₂|u^(n))=v₂ ^(n). For every w₁₁ and w₂₂,generate the codewords X₁ ^(n)(w₁₁|u^(n),v₁ ^(n)) and X₂^(n)(w₂₂|u^(n),v₂ ^(n)) independently according to the distributions

${P\left\{ {{X_{i}^{n}\left( {{w_{ii}u^{n}},v_{i}^{n}} \right)} = {x_{i}^{n}{E\left( {u^{n},v_{1}^{n},v_{2}^{n}} \right)}}} \right\}} = {\prod\limits_{t = 1}^{n}{P\left( {{x_{it}u_{t}},v_{it}} \right)}}$

for i=1,2. This completes our codebook construction.

Next, the encoding and decoding operations are described. SupposeW₁=(w₁₀,w_(1d),w₁₁) and W₂=(w₂₀,w_(2d),w₂₂). Encoders 1 and 2 send thepairs (w₁₀,w_(1d)) and (w₂₀,w_(2d)), respectively, to the cooperationfacilitator. Thus for i=1,2, φ_(i)(w_(i))=(w_(i0),w_(id)). Thecooperation facilitator then transmits

ψ₁(φ₁(w ₁),φ₂(w ₂))=(w ₂₀ ,Z ₁)

ψ₂(φ₁(w ₁),φ₂(w ₂))=(w ₁₀ ,Z ₂)

to encoders 1 and 2, respectively.

Using its knowledge of (w₁,w₂₀,Z₁), encoder 1 uses the (U^(n),V₁^(n))-codebook to transmit X₁ ^(n) (w₁₁|U^(n),V₁ ^(n)). Similarly, usingknowledge obtained from the cooperation facilitator, encoder 2 transmitsX₂ ^(n)(w₂₂|U^(n),V₂ ^(n)). It is worth noting that using thecooperation facilitator to transmit Z₁ and Z₂ is superior to simplyhaving one of the encoders act as the cooperation facilitator, becausethe encoders can receive Z₁ and Z₂ without either encoder incurring thepenalty in terms of loss of capacity associated with transmitting Z₁ andZ₂. That penalty is incurred by the cooperation facilitiator, which ischosen due to it having idle capacity.

The decoder uses joint typicality decoding. Upon receiving Y^(n) thedecoder looks for a unique pair (w₁,w₂) such that

(U ^(n)(w ₁₀ ,w ₂₀),V ₁ ^(n)(w _(1d) ,Z ₁),V ₂ ^(n)(w _(2d) ,Z ₂),

X ₁ ^(n)(w ₁₁),X ₂ ^(n)(w ₂₂),Y ^(n))∈A _(∈) ^((n)).  (5)

If such a (w₁,w₂) doesn't exist or exists but is not unique, the decoderdeclares an error. Although specific processes are described above forgenerating codes, as can readily be appreciated, other processes can beutilized to generate codes and code books that enable simpleimplementation of encoders and/or low latency encoding performance asappropriate to the requirements of specific applications in accordancewith various embodiments of the invention.

1.3 Error Analysis

In this section, the achievability scheme is studied more closely andsufficient conditions are provided for (R₁,R₂) such that the probabilityof error goes to zero. This immediately leads to Theorem 9 whichcharacterizes an achievable rate region for the MAC with transmittercooperation.

Suppose the message pair (w₁,w₂) is transmitted, wherew_(i)=(w_(i0),w_(id),w_(ii)). If (w₁,w₂) is the unique pair thatsatisfies Equation (5) then there is no error. If such a pair does notexist or is not unique, an error occurs. This event can be denoted by ∈.Since directly finding an upper bound on P(∈) is not straightforward. Ecan be upper bound by the union of a finite number of events and thenapply the union bound. Detailed proofs of the bounds mentioned in thissection are given in Parham Noorzad. Michelle Effros, and MichaelLangberg. On the Cost and Benefit of Cooperation (Extended Version),arxiv.org/abs/1504.04432, 17 Apr. 2015, which is hereby incorporated byreference in its entirety.

In what follows, denote U^(n)(w₁₀,w₂₀) and V_(i) ^(n)(W_(id),•|U^(n)) byU^(n) and V_(i) ^(n)(•), respectively. In addition, define

X _(i) ^(n)(•)=X _(i) ^(n)(w _(ii) |U ^(n) ,V _(i) ^(n)(•)).

Furthermore, denote instances of V_(i) ^(n)(•) and X_(i) ^(n)(•) withv_(i)(•) and χ_(i)(•), respectively. Also write V_(i) ^(n) and X_(i)^(n) instead of V_(i) ^(n)(w_(id),Z_(i)|U^(n)) and X_(i)^(n)(w_(ii)|U^(n),V_(i) ^(n)).

Denote the output of the decoder with (ŵ₁,ŵ₂). Denote U^(n)(ŵ₁₀,ŵ₂₀)with Û^(n) and similarly define {circumflex over (V)}_(i) ^(n) and{circumflex over (X)}_(i) ^(n) for i=1,2.

Next describe the error events. First, define ∈₀ as

∈₀={(U ^(n) ,V ₁ ^(n) ,V ₂ ^(n))∉A _(δ) ^((n))}.  (6)

When ∈₀ does not occur, the CF transmits (w₂₀,Z₁) and (w₁₀,Z₂) toencoders 1 and 2, respectively, which correspond to a jointly typicaltriple (U^(n),V₁ ^(n),V₂ ^(n)). Using Mutual Covering Lemma for weaklytypical sets is described in Parham Noorzad. Michelle Effros, andMichael Langberg, On the Cost and Benefit of Cooperation (ExtendedVersion), arxiv.org/abs/1504.04432, 17 Apr. 2015, which is herebyincorporated by reference in its entirety, it can be shown that P(∈₀)goes to zero if ζ>4∈, where ζ is defined by Equation (3).

Next, define ∈₁ as

∈₁={(U ^(n) ,V ₁ ^(n) ,V ₂ ^(n) ,X ₁ ^(n) ,X ₂ ^(n) ,Y ^(n))∉A _(∈)^((n))}.

This is the event where the codewords of the transmitted message pairare not jointly typical with the received output Y^(n). Then. P(∈₁\∈₀)→0as n→∞ if ζ<Θ(∈)−4δ.

If an error occurs and ∈₁ ^(c) holds, there must exist a message pair(ŵ₁,ŵ₂) different from (w₁,w₂) that satisfies (5). The message pair(ŵ₁,ŵ₂), where ŵ_(i)=(ŵ_(i0),ŵ_(id),ŵ_(ii)), may have(ŵ₁₀,ŵ₂₀)≠(w₁₀,w₂₀)) or (ŵ₁₀,ŵ₂₀)=(w₁₀,w₂₀).

Define ∈_(U) as the event where (ŵ₁₀,ŵ₂₀)≠(w₁₀,w₂₀). In this case,(Û^(n),{circumflex over (V)}₁ ^(n),{circumflex over (V)}₂^(n),{circumflex over (X)}₁ ^(n),{circumflex over (X)}₂ ^(n)) and Y^(n)are independent, which implies that P(∈_(U)) goes to zero ifR₁+R₂<I(X₁,X₂;Y)−ζ−7∈.

If (ŵ₁₀,ŵ₂₀)=(w₁₀,w₂₀), then either (ŵ_(1d),ŵ_(2d))≠(w_(1d),w_(2d)) or(ŵ_(1d),ŵ_(2d))=(w_(1d),w_(2d)). If (ŵ_(1d),ŵ_(2d))≠(w_(1d),w_(2d)),then ŵ_(1d)≠w_(1d) but ŵ_(2d)=w_(2d), or ŵ_(2d)≠w_(2d) butŵ_(1d)=w_(1d), or ŵ_(1d)≠w_(1d) and ŵ_(2d)≠w_(2d).

Let (i,j)∈{(1,2), (2,1)}. If ŵ_(id)≠w_(id) and ŵ_(jd)=w_(jd), it may beŵ_(jj)≠w_(jj) or ŵ_(jj)=w_(jj). Denote the former event by ∈_(V) _(i)_(X) _(j) , and the latter by ∈_(V) _(i) . Finally, denote the eventwhere ŵ_(1d)≠w_(1d) and ŵ_(2d)≠w_(2d) with ∈_(V) ₁ _(V) ₂ .

For (i,j)∈{(1,2),(2,1)}, when ∈_(V) _(i) _(X) _(j) occurs, ({circumflexover (V)}₁ ^(n),{circumflex over (V)}₂ ^(n),{circumflex over (X)}₁^(n),{circumflex over (X)}₂ ^(n)) and Y^(n) are independent given(U^(n),V_(j) ^(n)(•)). This implies P(∈_(V) _(i) _(X) _(j) )→0 if(R_(i)-R_(i0))+R_(jj)<I(X₁,X₂;Y|U,V_(j))−ζ−8∈.

For (i,j)∈{(1,2),(2,1)}, when ∈_(V) _(i) occurs, it can be shown that({circumflex over (V)}₁ ^(n),{circumflex over (V)}₂ ^(n),{circumflexover (X)}₁ ^(n),{circumflex over (X)}₂ ^(n)) and Y^(n) are independentgiven (U^(n),V_(j) ^(n)(•),X_(j) ^(n)(•)). This implies P(∈_(V) _(i) )→0if R_(i)-R_(i0)<I(X_(i);Y|U,V_(j),X_(j))−ζ−8∈.

If ∈_(V) _(i) _(X) _(j) occurs, ({circumflex over (V)}₁ ^(n),{circumflexover (V)}₂ ^(n),{circumflex over (X)}₁ ^(n),{circumflex over (X)}₂ ^(n))and Y^(n) are independent given U^(n). Thus P(∈_(V) ₁ _(V) ₂ ) goes tozero if

(R ₁-R ₁₀)+(R ₂-R ₂₀)<I(X ₁ ,X ₂ ;Y|U)−ζ−8∈.

Finally, if an error occurs and the message pairs have the same(w₁₀,w₂₀) and the same (w_(1d),w_(2d)), they must have different(w₁₁,w₂₂). Define the events ∈_(X) _(i) and ∈_(X) ₁ _(X) ₂ similarly tothe events for (w_(1d),w_(2d)). The relations

{circumflex over (X)} _(i) ^(n)→(U ^(n) ,V ₁ ^(n) ,V ₂ ^(n) ,X _(j)^(n))→Y ^(n)

({circumflex over (X)} ₁ ^(n) ,{circumflex over (X)} ₂ ^(n))→(U ^(n) ,V₁ ^(n) ,V ₂ ^(n))→Y ^(n),

hold for the events ∈_(X) _(i) and ∈_(X) ₁ _(X) ₂ , respectively. Fromthese relations it follows that

P(∈_(X) _(i) )→0 if R _(ii) <I(X _(i) ;Y|U,V ₁ ,V ₂ ,X _(j))−4∈.

P(∈_(X) ₁ _(X) ₂ )→0 if R ₁₁ +R ₂₂ <I(X ₁ ,X ₂ ;Y|U,V ₁ ,V ₂)−4∈.

Not surprisingly, these bounds closely resemble the bounds that appearin the capacity region of the classical MAC.

The bounds given in this section can be simplified further by replacingR_(i)-R_(i0) and R_(ii) with (R_(i)-C_(i0))⁺and (R_(i)-C_(i) ^(in))⁺,respectively, and noting that the set of all (x,y) that satisfy(x−a)⁺+(y−b)⁺<c is the same as the set of all (x,y) that satisfy x−a<c,y−b<c, and (x−a)+(y−b)<c.

Note that the general error event ∈ is a subset of the union of theerror events defined above. Thus if the union bound is applied and δ,∈,and ζ are chosen to be arbitrarily small, we obtain Theorem 9.

1.4 The Gaussian MAC

The Gaussian MAC is defined as the channel Y_(t)=X_(1t)+X_(2t)+Z_(t),where {Z_(t)}_(t=1) ^(n) is an i.i.d. Gaussian process independent of(X₁ ^(n),X₂ ^(n)) and each Z_(t) is a Gaussian random variable with meanzero and variance N. In addition, the output power of encoder i isconstrained by P_(i), that is, Σ_(t=1) ^(n)x_(it) ²≦nP_(i), where x_(it)is the output of encoder i at time t for i=1,2.

For the Gaussian MAC, the definition of an achievable rate pair can bemodified by adding the encoder power constraints to the definition ofthe (n,M₁,M₂) code for a MAC with a CF. Then the rate region

is achievable for the Gaussian MAC, where

is the same as

(Theorem 9) with the additional constraints

[X_(i) ²]≦P_(i) for i=1,2 on the input distributionP(u,v₁,v₂)P(x₁|u,v₁)P(x₂|u,v₂). This follows by replacing entropies withdifferential entropies and including the input power constraints in thedefinition of A∈^((n)). This is possible since weakly typical sets areused (rather than strongly typical sets) in the proof of Theorem 9.

If, in the calculation of

, we limit ourselves only to Gaussian input distributions, we get a rateregion which we denote by

. Note that

is an inner bound for the capacity region of a Gaussian MAC with a CF.Denote the signal to noise ratio of encoder i with

$\gamma_{i} = \frac{P_{i}}{N}$

and define γ=√{square root over (γ₁γ₂)}. The rate region

is given by the next theorem.

Theorem 4.

For the Gaussian MAC with a (C_(in),C_(out)) CF, the achievable rateregion

is given by the set of all rate pairs (R₁,R₂) that for {i,j}={1,2}satisfy

$\mspace{20mu} {R_{i} < {{\frac{1}{2}{\log \left( {1 + {\rho_{ii}^{2}\gamma_{i}}} \right)}} + C_{i}^{in}}}$$\mspace{20mu} {R_{i} < {{\frac{1}{2}{\log \left( {1 + {{\overset{\sim}{\rho}}_{ii}^{2}\gamma_{i}}} \right)}} + C_{i\; 0}}}$$\mspace{20mu} {{R_{1} + R_{2}} < {{\frac{1}{2}{\log \left( {1 + {\rho_{11}^{2}\gamma_{1}} + {\rho_{22}^{2}\gamma_{2}}} \right)}} + C_{1}^{in} + C_{2}^{in}}}$$\mspace{20mu} {{R_{1} + R_{2}} < {{\frac{1}{2}{\log \left( {1 + {\rho_{ii}^{2}\gamma_{i}} + {{\overset{\sim}{\rho}}_{jj}^{2}\gamma_{j}}} \right)}} + C_{i}^{in} + C_{j\; 0}}}$${R_{1} + R_{2}} < {{\frac{1}{2}{\log \left( {1 + {\left( {1 - \rho_{10}^{2}} \right)\gamma_{1}} + {\left( {1 - \rho_{20}^{2}} \right)\gamma_{2}} + {2\rho_{0}\rho_{1d}\rho_{2d}\overset{\_}{\gamma}}} \right)}} + C_{10} + C_{20}}$$\mspace{20mu} {{R_{1} + R_{2}} < {\frac{1}{2}{\log \left( {1 + \gamma_{1} + \gamma_{2} + {2\left( {{\rho_{10}\rho_{20}} + {\rho_{0}\rho_{1d}\rho_{2d}}} \right)\overset{\_}{\gamma}}} \right)}}}$

for some ρ₁₀,ρ₂₀,ρ_(1d),ρ_(2d)∈[0,1], and nonnegative constants C₁₀ andC₂₀ that satisfy Equation (1). In the above inequalities ρ₀,ρ_(ii), and{tilde over (ρ)}_(ii) (for i=1,2) are given by

$\begin{matrix}{{{\frac{1}{2}\log \frac{1}{1 - \rho_{0}^{2}}} \leq {\left( {C_{1}^{out} - C_{20}} \right) + \left( {C_{2}^{out} - C_{10}} \right)}}{\rho_{ii}^{2} = {1 - \rho_{i\; 0}^{2} - \rho_{i\; d}^{2}}}{{\overset{\sim}{\rho}}_{ii}^{2} = {1 - \rho_{i\; 0}^{2} - {\rho_{0}^{2}{\rho_{i\; d}^{2}.}}}}} & (7)\end{matrix}$

Theorem 4 is proved in Parham Noorzad. Michelle Effros, and MichaelLangberg. On the Cost and Benefit of Cooperation (Extended Version),arxiv.org/abs/1504.04432, 17 Apr. 2015, which is hereby incorporated byreference in its entirety.

Using Theorem 4, the maximum sum-rate of a scheme in accordance withmany embodiments of the invention can be calculated for the GaussianMAC. The “sum-rate gain” of a cooperation scheme can be defined as thedifference between the maximum sum-rate of that scheme and the maximumsum-rate of the classical MAC scheme. FIG. 8 illustrates the plot of themaximum sum-rate gain achieved by the described scheme for the GaussianMAC with γ₁=γ₂=10³ and C₁ ^(in)=C₂ ^(in)=0.2 as a function of C_(out).FIG. 8 plots the sum-rate gain of a scheme as a function of C₁ ^(out)=C₂^(out)=:C_(out) for γ₁=γ₂=10³, C₁ ^(in)=C₂ ^(in)=0.2 andC_(out)∈[0,0.25]. The conferencing bounds are also plotted in additionto the no conferencing sum-rate, which is the sum-rate corresponding toa scheme that splits the rate between the coordination and the classicalMAC strategies and does not make use of conferencing (C₁₀=C₂₀=0).

Note that for any value of C_(out) for which the gain in sum-rate isgreater than 4C_(out), adding a (C_(in),C_(out))−CF to the Gaussian MACresults in a network that does not satisfy the edge removal property.The reason is that if the output edges of the (C_(in),C_(out))−CF areremoved, the decrease in sum-capacity is greater than 4C_(out), whichimplies the decrease in either R₁ or R₂ (or both) is greater than2C_(out), which is the total capacity of the removed edges. On the plot,these are the points on our curve which fall above the “edge removalline”, that is, the line whose equation is given by gain=4C_(out).

As we see, the scheme that makes no use of conferencing performs wellwhen C_(out)<<C_(in), and the conferencing scheme works well whenC_(out) is close to C_(in) (and is optimal when C_(out)≧C_(in)). Thusboth strategies are necessary for our scheme to perform well over theentire range of C_(out). In this case study, the maximum sum-rate of

could have been obtained by a carefully designed time sharing betweenencoders which only cooperate through conferencing and encoders that useour scheme without conferencing.

2. Unbounded Benefit of Encoder Cooperation for K-User MACs

Cooperation strategies that allow communication devices to work togethercan improve network capacity. This section generalizes the “cooperationfacilitator” (CF) model from the 2-user to the k-user multiple accesschannel (MAC), extending capacity bounds, characterizing all k-user MACsfor which the sum-capacity gain of encoder cooperation exceeds thecapacity cost that enables it, and demonstrates an infinite benefit-costratio in the limit of small cost.

In the “MAC with CF” model introduced previously, a node called thecooperation facilitator (CF) helps a MAC's encoders to exchangeinformation before they transmit their codewords over the MAC. See FIG.9 which illustrates a network consisting of a k-user MAC and a CF. Thecapacity benefit of a CF in a 2-user MAC can far exceed the ratereceived by each encoder, and the set of all memoryless 2-user MACs forwhich encoder cooperation with a CF results in a large gain.

In this section, the results obtained with respect to a MAC with CF canbe generalized from the 2-user MAC to a k-user MAC. The k-user MAC witha CF provides a general setting for the study of cooperation amongmultiple encoders and captures previous cooperation models. Descriptionsof the model and main results follow.

Fix an integer k≧2. Consider a network consisting of a k-user MAC and a(C_(in),C_(out))−CF, as shown in FIG. 9. Here. W_([k])=(W₁, . . .,W_(k)) are the messages from the k encoders, X_([k]) ^(n)=(X₁ ^(n), . .. ,X_(k) ^(n)) are the corresponding channel inputs, Ŵ_([k])=(Ŵ₁ . . . .,Ŵ_(k)) are the message reproductions at the decoder, and vectorsC_(in),C_(out)∈

describe the capacities of the CF input and output

links. The notation [x] describes the set {1, . . . ,└x┘} for any realnumber x≧1.

In the first step of cooperation, each encoder sends a rate-limitedfunction of its message to the CF and the CF sends a rate-limitedfunction of what it receives back to each encoder. Communication betweenthe encoders and the CF continues for a finite number of rounds, witheach node potentially using information received in prior rounds todetermine its next transmission. Once the communication between the CFand the encoders is done, each encoder uses its message and what it haslearned through the CF to choose a codeword, which it transmits acrossthe channel.

The main result described further below determines the set of MACs wherethe benefit of encoder cooperation through a CF can grow very quicklywith Cot. Specifically, it can be shown that for any fixed C_(in)∈

and any MAC where the sum-capacity with full cooperation. (Full

cooperation means all encoders have access to all k messages.) exceedsthe sum-capacity without cooperation, the sum-capacity of that MAC witha (C_(in),C_(out))−CF has an infinite directional derivative atC_(out)=0 in every direction v∈

. A capacity region outer bound for the MAC can also be derived with a(C_(in),C_(out))−CF. The inner and outer bounds agree when the entriesof C_(out) are sufficiently larger than those of C_(in).

The achievability result are proved below using a coding scheme thatcombines single-step conferencing, coordination, and classical MACcoding. In conferencing, each encoder sends part of its message to allother encoders by passing that information through the CF. (Note it ispossible to handle encoders that send different parts of their messagesto different encoders.) The coordination strategy in various embodimentsof the invention, is a modified version of Marton's coding scheme forthe broadcast channel. The CF shares information with the encoders thatenables them to transmit codewords that are jointly typical with respectto a dependent distribution; this is proven using a multivariate versionof the covering lemma. The MAC strategy is Ulrey's extension ofAhlswede's and Liao's coding strategy to the k-user MAC.

A special case of this model with k=3 can be presented with respect tothe Gaussian MAC. In several embodiments of the present invention, itcan be shown that a single conferencing step is not optimal in general,even though it is optimal when k=2. Finally, outer bounds can be appliedfor the k-user MAC with a CF to obtain an outer bound for the k-user MACwith conferencing. The resulting outer bound is tight when k=2. Proofdetails for results relating to k-user MACs appear in Parham Noorzad.Michelle Effros, and Michael Langberg. The Unbounded Benefit of EncoderCooperation for the k-User MAC(Extended Version),arxiv.org/abs/1601.06113, 22 Jan. 2016, which is hereby incorporated byreference in its entirety.

2.1 K-User MACs Model and Results

Consider a network with k encoders, a CF, and a decoder as illustratedin FIG. 9. For each j∈[k], encoder j communicates with the CF usinglossless links of capacities C_(in) ^(j)≧0 and C_(out) ^(j)≧0 going toand from the CF, respectively. The k encoders communicate with thedecoder through a memoryless MAC

$\left( {{\prod\limits_{j = 1}^{k}\; _{j}},{p\left( {\left. y \middle| x_{1} \right.,\ldots \mspace{14mu},x_{k}} \right)},} \right).$

Each encoder j∈[k] wishes to transmit a message W_(j)∈[2 ^(nR) ^(j) ] tothe decoder. This is accomplished by first exchanging information withthe CF and then transmitting across the MAC. Communication with the CFoccurs in L steps. For each j∈[k] and l∈[L], sets (

)_(l=1) ^(L) and (

_(jl))_(l=1) ^(L), respectively, describe the alphabets of symbols thatencoder j can send to and receive from the CF in step l. These alphabetssatisfy the link capacity constraints Σ_(l=1) ^(L) log|

|≦nC_(in) ^(j) and Σ_(l=1) ^(L) log|

_(jl)|≦nC_(out) ^(j). The operation of encoder j and the CF,respectively, at step l are given by

ϕ_(j ):  [2^(nR_(j))] × _(j)^( − 1) → _(j )$\left. {\psi_{j\; }\text{:}\mspace{14mu} {\prod\limits_{i = 1}^{k}\; _{i}^{}}}\rightarrow{_{j\; }.} \right.$

where

=Π_(l′=1) ^(l)

and

=_(l′=1) ^(l)

. After its exchange with the CF, encoder j applies a function

ƒ_(j):[2^(nR) ^(j) ]×

→χ_(j) ^(n),

to choose a codeword, which it transmits across the channel. The decoderreceives channel output Y^(n) and applies

$\left. {g\text{:}\mspace{14mu} ^{n}}\rightarrow{\prod\limits_{j = 1}^{k}{\left\lbrack 2^{{nR}_{j}} \right\rbrack.}} \right.$

to obtain estimate Ŵ_([k]) of messages W_([k]).

The encoders, CF, and decoder together define a

((2^(nR) ¹ , . . . ,2^(nR) ^(k) ),n,L)

code for the MAC with a (C_(in),C_(out))−CF. The code's average errorprobability is P_(e) ^((n))=P{g(Y^(n))≠W_([k])}, where W_([k]) is arandom vector uniformly distributed on Π_(j=1) ^(k)[2^(nR) ^(j) ]. Arate vector R_([k])=(R₁, . . . ,R_(k)) is achievable if there exists asequence of ((2^(nR) ¹ , . . . ,2^(nR) ^(k) ),n,L) codes with P_(e)^((n))→0 as n→∞. The capacity region,

(C_(in),C_(out)), is defined as the closure of the set of all achievablerate vectors.

Using the coding scheme to be introduced below, an inner bound can beobtained for the capacity region of the k-user MAC with a(C_(in),C_(out))−CF. The following definitions are useful for describingthat bound. For every nonempty S⊂[k], define set χ_(S)=Π_(j∈S)χ_(j) withelements denoted by x_(S)=(x_(j))_(j∈S). Choose vectorsC₀=(C_(j0))_(j=1) ^(k) and C_(d)=(C_(jd))=_(j=1) ^(k) in

such that for all j∈[k],

$\begin{matrix}{C_{j\; 0} \leq C_{in}^{j}} & (8) \\{{C_{jd} + {\sum\limits_{i \neq j}C_{i\; 0}}} \leq {C_{out}^{j}.}} & (9)\end{matrix}$

Here C_(j0) is the number of bits per channel use encoder j sendsdirectly to the other encoders via the CF and C_(jd) is the number ofbits per channel use the CF transmits to encoder j to implement thecoordination strategy. Subscript “d” in C_(jd) alludes to the dependencecreated through coordination. Let S_(d)={j∈[k]:C_(jd)≠0} be the set ofencoders that participate in this dependence, and define

(S_(d)) to be the set of all distributions of the form

${p\left( u_{0} \right)} \cdot {\prod\limits_{i \in S_{d}^{c}}{{p\left( u_{i} \middle| u_{0} \right)} \cdot {p\left( {\left. u_{S_{d}} \middle| u_{0} \right.,u_{S_{d}^{c}}} \right)} \cdot {\prod\limits_{j \in {\lbrack k\rbrack}}{{p\left( {\left. x_{j} \middle| u_{0} \right.,u_{j}} \right)}.}}}}$

that satisfy ζ_(S)>0 for all S⊂S_(d), where

$\zeta_{S} = {{\sum\limits_{j \in S}C_{jd}} - {\sum\limits_{j \in S}{H\left( U_{j} \middle| U_{0} \right)}} + {{H\left( {\left. U_{S} \middle| U_{0} \right.,U_{S_{d}^{c}}} \right)}^{*}.}}$

For any C₀ and C_(d) satisfying Equations (8) and (9) and any p∈

(S_(d)), let

(C₀,C_(d),p) be the set of all R_([k]) that, for every S,T⊂[k], satisfy*The constraint on ζ_(S) is imposed by the multivariate covering lemma,which we use in the proof of our inner bound.

$\begin{matrix}{{{\sum\limits_{j \in A}\left( {R_{j} - C_{j\; 0}} \right)^{+}} + {\sum\limits_{j \in {B\bigcap T}}\left( {R_{j} - C_{in}^{j}} \right)^{+}}} < {{I\left( {U_{A},{X_{A\bigcup{({B\bigcap T})}};\left. Y \middle| U_{0} \right.},Y_{B},X_{B\backslash T}} \right)} - \zeta_{{({A\bigcup B})}\bigcap S_{d}}}} & (10)\end{matrix}$

for some sets A and B for which S∩S_(d) ^(c) ⊂A⊂S and S^(c)∩S_(d) ^(c)⊂B⊂S^(c), in addition to

$\begin{matrix}{{\sum\limits_{j \in {\lbrack k\rbrack}}R_{j}} < {{I\left( {X_{\lbrack k\rbrack};Y} \right)} - {\zeta_{S_{d}}.}}} & (11)\end{matrix}$

Here U₀ encodes the “common message,” which contains nC_(j0) bits fromeach W_(j) and is shared with all other encoders through the CF; eachrandom variable U_(j) captures the information encoder j receives fromthe CF to create dependence with the codewords of other encoders.

Next, the inner bound for the k-user MAC can be stated with encodercooperation via a CF. The coding strategy that achieves this inner boundonly uses a single step of cooperation.

Theorem 5 (Inner Bound).

For any MAC (χ_([k]),p(y|x_([k])),ψ) with a (C_(in),C_(out))−CF,

(C _(in) ,C _(out))⊃ ∪

(C ₀ ,C _(d) ,p)

where Ā denotes the closure of set A and the union is over all C₀ andC_(d) satisfying (8), (9), and p∈

(S_(d)).

The region given in Theorem 5 is convex and thus does not require theconvex hull operation. We can prove this by applying the same techniqueused for the 2-user MAC.

In the above theorem, if for every S,T⊂[k] with S∪T≠, we choose A=S andB=S^(c), then our region simplifies to the set of all rate vectorssatisfying

${{{\sum\limits_{j \in S}\left( {R_{j} - C_{j\; 0}} \right)^{+}} + {\sum\limits_{j \in {T\backslash S}}\left( {R_{j} - C_{in}^{j}} \right)^{+}}} < {{I\left( {{X_{S\bigcup T};\left. Y \middle| U_{0} \right.},U_{S^{c}},X_{S^{c}\bigcap T^{c}}} \right)} - \zeta_{S_{d}}}},$

in addition to Equation (11) for C₀ and C_(d) (satisfying Equations (8)and (9)) and some distribution p∈

(S_(d)).

Corollary 1 treats the case where the CF transmits the bits it receivesfrom each encoder to all other encoders without change. We obtain thisresult from Theorem 5 by setting C_(jd)=0 and X_(j)=U_(j) for all j∈[k]and choosing A=S and B=S^(c) for every S,T⊂[k].

Corollary 1 (Forwarding Inner Bound).

For any MAC (χ_([k]),p(y|x_([k])),ψ),

(C_(in),C_(out)) contains the set of all rate vectors R_([k]) that forsome constants (C_(j0))_(j∈[k]) (satisfying Equations (8) and (9) withC_(jd)=0 for all j) and some distribution p(u₀)Π_(j=1) ^(k)p(x_(j)|u₀),satisfy

${{\sum\limits_{j \in S}R_{j}} < {{I\left( {{X_{S};\left. Y \middle| U_{0} \right.},X_{S^{c}}} \right)} + {\sum\limits_{j \in S}C_{j\; 0}}}},$

for every nonempty S⊂[k], and

${\sum\limits_{j \in {\lbrack k\rbrack}}R_{j}} < {{I\left( {X_{\lbrack k\rbrack};Y} \right)}.}$

As stated above, it is important to determine when the benefit ofcooperation is in some sense large. Here we measure the benefit ofcooperation by comparing the gain in sum-capacity to the number of bitsshared with the encoders to enable that gain.

For any MAC (χ_([k]),p(y|x_([k])),ψ) with a (C^(in),C^(out))−CF, definethe sum-capacity as

$C_{sum} = {{\sum\limits_{j = 1}^{k}{R_{j}.}}}$

For a fixed C_(in)∈

, define the “sum-capacity gain” G:

→

as

G(C _(out))=C _(sum)(C ^(in) ,C ^(out))−C _(sum)(C ^(in),0),

where C_(out)=(C_(out) ^(j))_(j=1) ^(k) and 0 is the all-zeros vector.Note that when C_(out)=0, no cooperation is possible, thus

${C_{sum}\left( {C^{in},0} \right)} = {\max\limits_{{p{(x_{1})}}\mspace{14mu} \ldots \mspace{14mu} {p{(x_{k})}}}{{I\left( {X_{\lbrack k\rbrack};Y} \right)}.}}$

Using these definitions, the main result of k-user MACs in accordancewith several embodiments of the invention is stated below, the proof ofwhich is given in Parham Noorzad. Michelle Effros, and Michael Langberg,The Unbounded Benefit of Encoder Cooperation for the k-User MAC(ExtendedVersion), arxiv.org/abs/1601.06113, 22 Jan. 2016, which is herebyincorporated by reference in its entirety.

Theorem 6 (Sum-Capacity).

Consider a discrete MAC (χ_([k]),p(y|x_([k])),ψ). Fix C_(in)∈

. Then the channel satisfies

(D _(v) G)(0)=∞

if and only if

${{\max\limits_{p{(x_{\lbrack k\rbrack})}}{I\left( {X_{\lbrack k\rbrack};Y} \right)}} > {\max\limits_{{p{(x_{1})}}\mspace{14mu} \ldots \mspace{14mu} {p{(x_{k})}}}{I\left( {X_{\lbrack k\rbrack};Y} \right)}}},$

where v∈

is any unit vector and D_(v)G is the directional derivative of G in thedirection of v.

While only a single step of cooperation is utilized in our achievabilityresult in (Theorem 5), the outer bound applies to coding schemes thatmake use of more than one step.

Theorem 7 (Outer Bound).

For the MAC (χ_([k]),p(y|x_([k])),ψ),

(C_(in),C_(out)) is a subset of the set of rate vectors R_([k]) that forsome distribution p(u₀)Π_(j=1) ^(k)p(x_(j)|u₀) satisfy

$\begin{matrix}{{\sum\limits_{j \in S}R_{j}} \leq {{I\left( {{X_{S};{YU_{0}}},{X_{S}c}} \right)} + {\sum\limits_{j \in S}C_{i\; n}^{j}}}} & (12)\end{matrix}$

for all ≠S⊂[k], in addition to

$\begin{matrix}{{\sum\limits_{j \in {\lbrack k\rbrack}}R_{j}} \leq {{I\left( {X_{\lbrack k\rbrack};Y} \right)}.}} & (13)\end{matrix}$

The proof of this theorem is given in Parham Noorzad. Michelle Effros,and Michael Langberg, The Unbounded Benefit of Encoder Cooperation forthe k-User MAC (Extended Version), arxiv.org/abs/1601.06113, 22 Jan.2016, which is hereby incorporated by reference in its entirety.

If the capacities of the CF output links are sufficiently large, theinner and outer bounds coincide and the capacity region is obtained.This follows by setting C_(j0)=C_(in) ^(j) for all j∈[k] in theforwarding inner bound (Corollary 1) and comparing it with the outerbound given in Theorem 7.

Corollary 2.

For the memoryless MAC (χ_([k]),p(y|x_([k])),ψ) with a(C_(in),C_(out))−CF, if for every j∈[k], we have

${C_{out}^{j} \geq {\sum\limits_{i:{i \neq j}}C_{i\; n}^{i}}},$

then our inner and outer bounds agree.

2.2 K-User MACs Coding Scheme

Choose nonnegative constants (C_(j0))_(j=1) ^(k) and (C_(jd))_(j=1) ^(k)such that for all j∈[k], (8) and (9) hold. Fix a distribution p∈

(S_(d)) and choose ∈,δ>0. Let

R _(j0)=min{R _(j) ,C _(j0)}

R _(jd)=min{R _(j) ,C _(in) ^(j) }−R _(j0)

R _(jj) =R _(j) −R _(j0) −R _(jd)=(R _(j) −C _(in) ^(j))⁺,

where x⁺=max{x,0} for any real number x. For every j∈[k], split themessage of encoder j as w_(j)=(w_(j0),w_(jd),w_(jj)) wherew_(j0)∈[2^(nR) ^(j0) ], w_(jd)∈[2^(nR) ^(jd) ], w_(jj)∈[2 ^(nR) ^(jj) ].Encoder j sends w_(j0) to the other encoders via the CF to implement thesingle-step conferencing strategy. In addition to w_(j0), encoder jsends w_(jd) to the CF to implement the coordination strategy. Note thatthe sum R_(j0)+R_(jd) is always less than or equal to C_(in) ^(j).Finally, encoder j sends the remaining part of its message, w_(jj), overthe channel using the classical MAC strategy.

Let

=Π_(j=1) ^(k)[2^(nR) ^(j0) ]. For every w₀∈

, let U₀ ^(n)(w₀) be distributed as

${P\left\{ {{U_{0}^{n}\left( w_{0} \right)} = u_{0}^{n}} \right\}} = {\prod\limits_{t = 1}^{n}{{p\left( u_{0t} \right)}.}}$

Given U₀ ^(n)(w₀)=u₀ ^(n), for every j∈[k], w_(jd)∈[2 ^(nR) ^(jd) ], andz_(j)∈[2 ^(nC) ^(jd) ], let U_(j) ^(n)(w_(jd),z_(j)|u₀ ^(n)) havedistribution

$\begin{matrix}{{P\left\{ {{U_{j}^{n}\left( {{w_{{jd},}z_{j}}u_{0}^{n}} \right)} = {{u_{j}^{n}{U_{0}^{n}\left( w_{0} \right)}} = u_{0}^{n}}} \right\}} = {\prod\limits_{t = 1}^{n}{{p\left( {u_{jt}u_{0t}} \right)}.}}} & (14)\end{matrix}$

For every (w₁, . . . ,w_(k)), define E(u₀ ^(n),μ₁, . . . ,μ_(k)) as theevent where U₀ ^(n)(w₀)=u₀ ^(n) and for every j∈[k],

U _(j) ^(n)(w _(jd) ,•|u ₀)=μ_(j)(•),  (15)

where μ_(j) is a mapping from [2 ^(nC) ^(jd) ] to

_(j) ^(n). Let

(u₀ ^(n),μ_([k])) be the set of all z_([k]) such that

(u ₀ ^(n),μ_([k])(z _([k])))∈A _(δ) ^((n))(U ₀ ,U _([k])),  (16)

where μ_([k])(z_([k]))=(μ₁(z₁), . . . ,μ_(k)(z_(k))) and A_(δ)^((n))(U₀,U_([k])) is the weakly typical set with respect to thedistribution p(u₀,u_([k])). If

(u₀ ^(n),μ_([k])) is empty, set Z_(j)=1 for all j∈[k]. Otherwise, letthe k-tuple Z_([k]) have joint distribution

$\begin{matrix}{{P\left\{ {{\forall{j \in {\lbrack k\rbrack \text{:}Z_{j}}}} = {z_{j}{E\left( {u_{0}^{n},\mu_{\lbrack k\rbrack}} \right)}}} \right\}} = {\frac{1}{{\left( {u_{0}^{n},\mu_{\lbrack k\rbrack}} \right)}}.}} & \;\end{matrix}$

Finally, given U₀ ^(n)(w₀)=u₀ ^(n) and U_(j) ^(n)(w_(jd),Z_(j))=u_(j)^(n), let X_(j) ^(n)(w_(jj)|u₀ ^(n),u_(j) ^(n)) be a random vector withdistribution

${P\left\{ {{{X_{j}^{n}\left( {{w_{jj}u_{0}^{n}},u_{j}^{n}} \right)} = {{x_{j}^{n}{U_{0}^{n}\left( w_{0} \right)}} = u_{0}^{n}}},{{U_{j}^{n}\left( {w_{jd},Z_{j}} \right)} = u_{j}^{n}}} \right\}} = {{p\left( {{x_{j}^{n}u_{0}^{n}},u_{j}^{n}} \right)} = {\prod\limits_{j = 1}^{n}{{p\left( {{x_{jt}u_{0t}},u_{jt}} \right)}.}}}$

The encoding and decoding processes is described next.

Encoding.

For every j∈[k], encoder j sends the pair (w_(j0),w_(jd)) to the CF. TheCF then sends ((w_(i0))_(i≠j),Z_(j)) back to encoder j. Encoder j, nowhaving access to w₀ and Z_(j), transmits X_(j) ^(n)(w_(jj)|U₀^(n)(w₀),U_(j) ^(n)(w_(jd),Z_(j))) over the channel.

Decoding.

The decoder, upon receiving Y^(n), maps Y^(n) to the unique k-tupleŵ_([k]) such that

(U ₀ ^(n)(ŵ ₀),(U _(j) ^(n)(ŵ _(jd) ,{circumflex over (Z)} _(j) |U ₀^(n)))_(j),(X _(j) ^(n)(ŵ _(jj) |U ₀ ^(n) ,U _(j) ^(n)))_(j) ,Y ^(n))∈A_(∈) ^((n)))(U ₀ ,U _([k]) ,X _([k]) ,Y).  (17)

If such a k-tuple does not exist, the decoder sets its output to thek-tuple (1, 1, . . . ,1).

2.3 Case Study: 2-User MAC

As noted above, when k=2, the achievability region in (Theorem 5)contains the region presented above. Here it can be shown that for thenetwork consisting of the 2-user Gaussian MAC with a((∞,∞),(C_(out),C_(out)))−CF, the region described above strictlycontains the region for Gaussian input distributions.

Theorem 5 implies that the capacity region of the mentioned networkcontains the set of all rate pairs R_([2]) that satisfy

R ₁≦max{I(X ₁ ;Y|U)−C _(1d) ,I(X ₁ ;Y|X ₂ ,U)}+C ₁₀

R ₂≦max{I(X ₂ ;Y|U)−C _(2d) ,I(X ₂ ;Y|X ₁ ,U)}+C ₂₀

R ₁ +R ₂ ≦I(X ₁ ,X ₂ ;Y|U)+C ₁₀ +C ₂₀

R ₁ +R ₂ ≦I(X ₁ ,X ₂ ;Y)

for some nonnegative constants C₁₀,C₂₀≦C_(out),

C _(1d) =C _(out) −C ₂₀

C _(2d) =C _(out) −C ₁₀,

and some distribution p(u)p(x₁,x₂|u) that satisfies

[X_(i) ²]≦P_(i) for i∈{1,2} and

I(X ₁ ;X ₂ |U)≦2C _(out) −C ₁₀ −C ₂₀.

When this region is calculated for the Gaussian MAC using a Gaussianinput distribution, we get (set γ_(i)=P_(i)/N for i∈{1,2} and γ=√{squareroot over (γ₁γ₂)}) the set consisting of all rate pairs satisfying

$R_{1} \leq {\max \left\{ {{{\frac{1}{2}\log \frac{1 + {\left( {1 - \rho_{1}^{2}} \right)\gamma_{1}} + {\left( {1 - \rho_{2}^{2}} \right)\gamma_{2}} + {2\; \rho_{0}\overset{\_}{\gamma}\sqrt{{\left( {1 - \rho_{1}^{2}} \right)\left( {1 - \rho_{2}^{2}} \right)}\;}}}{1 + {\left( {1 - \rho_{0}^{2}} \right)\left( {1 - \rho_{2}^{2}} \right)\gamma_{2}}}} - C_{1d}},{{{\frac{1}{2}{\log\left( {1 + {\left( {1 - \rho_{0}^{2}} \right)\left( {1 - {\rho_{1}^{2}\gamma_{1}}} \right)}} \right\}}} + {C_{10}R_{2}}} \leq {{\max \left\{ {{{\frac{1}{2}\log \frac{1 + {\left( {1 - \rho_{1}^{2}} \right)\gamma_{1}} + {\left( {1 - \rho_{2}^{2}} \right)\gamma_{2}} + {2\; \rho_{0}\overset{\_}{\gamma}\sqrt{\left( {1 - \rho_{1}^{2}} \right)\left( {1 - \rho_{2}^{2}} \right)}}}{1 + {\left( {1 - \rho_{0}^{2}} \right)\left( {1 - \rho_{1}^{2}} \right)\gamma_{1}}}} - C_{2d}},{\frac{1}{2}{\log \left( {1 + {\left( {1 - \rho_{0}^{2}} \right)\left( {1 - \rho_{2}^{2}} \right)\gamma_{2}}} \right)}}} \right\}} + {C_{20}\mspace{79mu} {and}R_{1}} + R_{2}} \leq {{\frac{1}{2}{\log \left( {1 + {\left( {1 - \rho_{1}^{2}} \right)\gamma_{1}} + {\left( {1 - \rho_{2}^{2}} \right)\gamma_{2}} + {2\rho_{0}\overset{\_}{\gamma}\sqrt{\left( {1 - \rho_{1}^{2}} \right)\left( {1 - \rho_{2}^{2}} \right)}}} \right)}\mspace{20mu} R_{1}} + R_{2}} \leq {\frac{1}{2}{\log \left( {1 + \gamma_{1} + \gamma_{2} + {2\rho_{0}\overset{\_}{\gamma}}} \right)}}}} \right.}$

for some ρ₁,ρ₂∈[0,1] and 0≦ρ₀≦√{square root over (1−2^(−2(C) ^(1d) ^(+C)^(2d) ⁾)}. FIG. 10 illustrates a comparison of the achievable sum-rategain for the Gaussian MAC with Gaussian input distribution using resultsdescribed above with respect to the Gaussian MAC (old) and resultsdescribed in this section (new). The difference is clear from the plot.2.4 The k-User Mac with Conferencing Encoders

Willems' conferencing model can be extended to the k-user MAC asfollows. Consider a k-user MAC where for every i,j∈[k] (in this section,i≠j by assumption), there is a link of capacity C_(ij)≧0 from encoder ito encoder j and a link of capacity C_(ji)≧0 back. See FIG. 11 whichillustrates in the k-user MAC with conferencing, for every i,j∈[k],there are links of capacities C_(ij) and C_(ji) connecting encoder i andencoder j.

As in 2-user conferencing, conferencing occurs over a finite number ofsteps. In the first step, for every j∈[k], encoder j transmits someinformation to encoder i (for every i with C_(ji)>0) that is a functionof its own message W_(j)∈[2 ^(nR) ^(j) ]. In each subsequent step, everyencoder transmits information that is a function of its message andinformation it learns before that step. Once the conferencing is over,each encoder transmits its codeword over the k-user MAC. Next define a((2^(nR) ¹ , . . . ,2^(nR) ^(k) ),n,L)-code for the k-user MAC with anL-step (C_(ij))_(i,j=1) ^(k)-conference. For every i,j∈[k] and l∈[L],fix a set

so that for every i,j∈[k],Σ_(l=1) ^(L) log|

|≦nC_(ij). Here

represents the alphabet of the symbol encoder i sends to encoder j instep l of the conference. For every l∈[L], define

=Π_(l′=1) ^(l)

. For j∈[k] encoder j is represented by the collection of functions(ƒ_(j),(h_(ji) ^((l)))_(i,l)) where

$\left. {f_{j}{\text{:}\mspace{14mu}\left\lbrack 2^{{nR}_{j}} \right\rbrack} \times {\prod\limits_{i:{i \neq j}}_{ij}^{L}}}\rightarrow ^{n} \right.$$\left. {h_{ji}^{(l)}{\text{:}\mspace{14mu}\left\lbrack 2^{{nR}_{j}} \right\rbrack} \times {\prod\limits_{i^{\prime}:{i^{\prime} \neq j}}_{i^{\prime}j}^{l - 1}}}\rightarrow _{ji}^{(l)} \right.$

The decoder is defined as g:ψ^(n) Π_(j=1) ^(k)[2^(nR) ^(j) ]. Thedefinition of an achievable rate vector and the capacity region aresimilar to those described above.

The next result compares the capacity region of a MAC with cooperationunder the conferencing and CF models. The proof is given in ParhamNoorzad. Michelle Effros, and Michael Langberg, The Unbounded Benefit ofEncoder Cooperation for the k-User MAC (Extended Version),arxiv.org/abs/1601.06113, 22 Jan. 2016, which is hereby incorporated byreference in its entirety.

Theorem 8.

The capacity region of a MAC with an L-step (C_(ij))_(i,j=1)^(k)-conference is a subset of the capacity region of the same MAC withan L-step cooperation via a (C_(in),C_(out))−CF if for all j∈[k],

$C_{i\; n}^{j} \geq {\sum\limits_{i:{i \neq j}}{C_{ji}\mspace{14mu} {and}\mspace{14mu} C_{out}^{j}}} \geq {\sum\limits_{i:{i \neq j}}{C_{ij}.}}$

Similarly) for every L, the capacity region of a MAC with L-stepcooperation via a (C_(in),C_(out))−CF is a subset of the capacity regionof the same MAC with a single-step (C_(ij))_(i,j=1) ^(k)-conference iffor all i,j∈[k],C_(ij)≧C_(in) ^(i).

Combining the first part of Theorem 8 with the outer bound from Theorem7 results in the next corollary.

Corollary 3 (Conferencing Outer Bound).

For the memoryless MAC (χ_([k]),p(y|x_([k])), ψ) with a (C_(ij))_(i,j=1)^(k)-conference, the set of achievable rate vectors is a subset of theset of rate vectors R_([k])that for some distribution p(u)Π_(j=1)^(k)p(x_(j)|u) satisfy

${\sum\limits_{j \in S}R_{j}} \leq {{I\left( {{X_{S};{YU}},{X_{S}c}} \right)} + {\sum\limits_{j \in S}{\sum\limits_{i \neq j}C_{ji}}}}$

for every nonempty S⊂[k], in addition to

${\sum\limits_{j \in {\lbrack k\rbrack}}R_{j}} \leq {{I\left( {X_{\lbrack k\rbrack};Y} \right)}.}$

While k-user conferencing is a direct extension of 2-user conferencing,there is nonetheless a major difference when k≧3. While it is well knownthat in the 2-user case a single conferencing step suffices to achievethe capacity region, the same is not true when k≧3, as is illustratednext.

A special case of this model for the 3-user Gaussian MAC is shown inFIG. 12A, which illustrates a conferencing structure. FIG. 12Billustrates an alternative conferencing structure where increasing thenumber of steps results in a larger capacity region. Consider thenetwork consisting of a 3-user MAC with conferencing. Fix positiveconstants C_(in) ¹ and C_(in) ². Let C₁₃=C_(in) ¹,C₂₃=C_(in) ²,C₃₁=C₃₂=C_(out) for C_(out)∈

_(≧0), and C₁₂=C₂₁=0.

Let

(C_(out)) and

(C_(out)) denote the capacity region of this network with one and twosteps of conferencing, respectively. For each L∈{1,2}, define thefunction g_(L)(C_(out)) as

g_(L)(C_(out)) = (R₁ + R₂).

Note that when L=1, g₁(C_(out))=g₁(0) for all C_(out) since nocooperation is possible when encoder 3 is transmitting at rate zero. Onthe other hand, as shown next, at least for some MACs (including theGaussian MAC), g′₂(0)=∞, that is, g₂ has an infinite slope at C_(out)=0.Note that

${g_{2}(0)} = {{g_{1}(0)} = {\max\limits_{{{p{(x_{1})}}{p{(x_{2})}}},x_{3}}{{I\left( {X_{1},{X_{2};{{YX_{3}} = x_{3}}}} \right)}.}}}$

Let p*(x₁)p*(x₂) and x*₃ achieve this maximum. If a MAC satisfies thecondition

${{\max\limits_{p{({x_{1},x_{2}})}}{I\left( {X_{1},{X_{2};{\left. Y \middle| X_{3} \right. = x_{3}^{*}}}} \right)}} > {g_{2}(0)}},$

then by Theorem 6 (for k=2), g′₂(0)=∞. Since g₁ is constant for allC_(out), while g₂ has an infinite slope at C_(out)=0, and g₁(0)=g₂(0),the 2-step conferencing region is strictly larger than the single-stepconferencing region. Using the same technique, a similar result for anyk≧3 can be shown; that is, there exist k-user MACs where the two-stepconferencing region strictly contains the single-step region.

3. Cooperation Increasing Network Reliability

In network cooperation strategies, nodes work together with the aim ofincreasing transmission rates or reliability. This section demonstratesthat enabling cooperation between the transmitters of a two-usermultiple access channel via a cooperation facilitator that has access toboth messages, always results in a network whose maximal- andaverage-error sum-capacities are the same—even when the informationshared with the encoders is negligible. Thus, for a multiple accesschannel whose maximal- and average-error sum-capacities differ, themaximal-error sumcapacity is not continuous with respect to the outputedge capacities of the facilitator. This shows that for some networks,sharing even a negligible number of bits per channel use with theencoders can yield a non-negligible benefit.

Cooperative strategies enable an array of code performance improvements,including higher transmission rates and higher reliability. In thiswork, the same benefit can be viewed as either an improvement in ratefor a given reliability or an improvement in reliability for a givenrate. A discussion of the latter perspective is included first.

Consider a network with multiple transmitters and a single receiver.Given a code, one can calculate the probability of error at the receiverfor each possible message vector. The probability of error, viewed as afunction of the transmitted message vector, provides a measure of thereliability of the code. The average- and maximal-error probabilities ofthe code are the average and maximum of the range of this function,respectively. To understand the relationship between cooperation andreliability, study how cooperation can be used to increase thereliability of a code. Specifically, seek to modify a code that achievessmall average error without cooperation to obtain a code at the samerate that achieves small maximal error using rate-limited cooperation.

To make this discussion more concrete, consider a network consisting ofa multiple access channel (MAC) and a cooperation facilitator (CF), asshown in FIG. 13. FIG. 13 illustrates a network consisting of amemoryless MAC and a CF. The CF is a node that can send and receivelimited information to and from each encoder. Each encoder, prior totransmitting its codeword over the channel, sends some information tothe CF. The CF then replies to each encoder over its output links. Thiscommunication may continue for a finite number of steps. The totalnumber of bits transmitted on each CF link is bounded by the product ofthe blocklength, n, and the capacity of that link. Once the encoders'communication with the CF is over, each encoder transmits its codewordover n uses of the channel.

In order to quantify the benefit of rate-limited cooperation in theabove network, a spectrum of error probabilities is defined that rangefrom average error to maximal error. The main result, described below,states that if for i∈{1,2}C_(in) ^(i) is increased (the capacity of thelink from encoder i to the CF) by some constant value and C_(out) ^(i)(the capacity of the link from the CF to encoder i) by any arbitrarilysmall amount, then any rate pair that is achievable in the originalnetwork under average error is achievable in the new network under astricter notion of error. This result, stated formally below, quantifiesthe relationship between cooperation and reliability. The proof, asillustrated in Parham Noorzad, Michelle Effros, and Michael Langberg.Can Negligible Cooperation Increase Network Reliability? (ExtendedVersion), arxiv.org/abs/1601.05769, 21 Jan. 2016, which is herebyincorporated by reference in its entirety, shows that the average- andmaximal-error capacity regions of the discrete memoryless broadcastchannel are identical.

A specific instance of an embodiment of the present invention is thecase where C_(in) ¹ and C_(in) ² are sufficiently large so that the CFhas access to both source messages. In this case, it can be shown thatwhenever C_(out) ¹ and C_(out) ² are strictly positive, themaximal-error capacity region of the resulting network is identical toits average-error capacity region. Applying this result to Dueck's“Contraction MAC.” which has a maximal-error capacity region strictlysmaller than its average-error capacity region, yields a network whosemaximal-error sum-capacity is not continuous with respect to thecapacities of its edges. The discontinuity in sum-capacity observed hereis related to the edge removal problem, which is discussed next.

The edge removal problem studies the change in network capacity thatresults from removing an edge of finite capacity. One instance of thisproblem considers removed edges of “negligible capacity.” Intuitively,an edge can be thought of as having negligible capacity if the number ofbits that it can carry in n channel uses grows sublinearly in n; forexample, an edge that can carry log n bits in n channel uses hasnegligible capacity. In this context, the edge removal problem askswhether removing an edge with negligible capacity from a network has anyeffect on the capacity region of that network. This result showing theexistence of a network with a discontinuous maximal-error sum-capacitydemonstrates the existence of a network where removing an edge withnegligible capacity has a non-negligible effect on its maximal-errorcapacity region.

Given that one may view feedback as a form of cooperation, similarquestions may be posed about feedback and reliability. It can be shownthat for some MACs, the maximal-error capacity region with feedback isstrictly contained in the average-error region without feedback. This isin contrast to various embodiments of the present invention of encodercooperation via a CF that has access to both messages and output edgesof negligible capacity. It is shown below that for any MAC with thistype of encoder cooperation, the maximal-error and average-error regionsare identical. Hence, unlike cooperation via feedback, the maximal-errorregion of a MAC with negligible encoder cooperation contains theaverage-error region of the same MAC without encoder cooperation.

In the next section, the model for increasing network reliability isintroduced. This is followed by a discussion of results. The proofs ofall theorems are available in Parham Noorzad. Michelle Effros, andMichael Langberg, Can Negligible Cooperation Increase NetworkReliability? (Extended Version), arxiv.org/abs/1601.05769, 21 Jan. 2016,which is hereby incorporated by reference in its entirety.

3.1 Reliability Models

Consider a network comprising two encoders, a (C_(in),C_(out))−CF, amemoryless MAC

(χ₁×χ₂ ,p(y|x ₁ ,x ₂),ψ),

and a decoder. Define an (n,M₁,M₂,J)-code for this network withtransmitter cooperation. For every real number x≧1, let [x] denote theset {1, . . . ,└x┘}, where └x┘ denotes the integer part of x. For eachi∈{1,2} fix two sequences of sets (

)_(j=1) ^(J) L and (

)_(j=1) ^(J) such that

${\log {_{i}^{J}}} = {{\sum\limits_{j = 1}^{J}{\log {_{ij}}}} \leq {nC}_{i\; n}^{i}}$${{\log {_{i}^{J}}} = {{\sum\limits_{j = 1}^{J}{\log {_{ij}}}} \leq {nC}_{out}^{i}}},$

where for all j∈[J],

$_{i}^{j} = {\prod\limits_{ = 1}^{j}_{i\; }}$${_{i}^{j} = {\prod\limits_{ = 1}^{j}_{i\; }}},$

and log denotes the natural logarithm. Here

represents the alphabet for the j^(th)-step transmission

from encoder i to the CF while

represents the alphabet for the j^(th)I-step transmission from the

CF to encoder i. The given alphabet size constraints are chosen to matchthe total rate constraints nC_(in) ^(i) and nC_(out) ^(i) over J stepsof communication between the two encoders and n uses of the channel. Fori∈{1,2}, encoder i is represented by ((φ_(ij))_(j=1) ^(J),ƒ_(i)), where

φ_(ij) :[M _(i)]×

→

captures the step-j transmission from encoder i to the CF, and

ƒ_(i) :[M _(i)]×

→χ_(i) ^(n).

captures encoder i's transmission across the channel. The CF isrepresented by the functions ((ψ_(1j))_(j=1) ^(J),(ψ_(2j))_(j=1) ^(J)),where for i∈{1,2} and j∈[J],

ψ_(ij):

×

→

captures the step-j transmission from the CF to encoder i. For eachmessage pair (m₁,m₂), i∈{1,2}, and j∈[J], define

u _(ij)=φ_(ij)(m _(i) ,v _(i) ^(j−1)

v _(ij)=ψ_(ij)(u ₁ ^(i) ,u ₂ ^(j)).

At step j, encoder i sends u_(ij) to the CF and receives v_(ij) from theCF. After the J-step communication between the encoders and the CF isover, encoder i transmits ƒ_(i)(m_(i),v_(i) ^(j)) over the channel. Thedecoder is represented by the function

g:ψ ^(n) →[M ₁ ]×[M ₂].

The probability that a message pair (m₁,m₂) is decoded incorrectly isgiven by

${\lambda_{n}\left( {m_{1},m_{2}} \right)} = {\sum\limits_{y^{n} \notin {g^{- 1}{({m_{1},m_{2}})}}}{{p\left( {\left. y^{n} \middle| {f_{1}\left( {m_{1},v_{1}^{J}} \right)} \right.,{f_{2}\left( {m_{2},v_{2}^{J}} \right)}} \right)}.}}$

The average probability of error, P_(e,avg) ^((n)), and the maximalprobability of error, P_(e,max) ^((n)), are defined as

$P_{e,{avg}}^{(n)} = {\frac{1}{M_{1}M_{2}}{\sum\limits_{m_{1},m_{2}}{\lambda_{n}\left( {m_{1},m_{2}} \right)}}}$${P_{e,{{ma}\; x}}^{(n)} = {\max\limits_{m_{1},m_{2\;}}{\lambda_{n}\left( {m_{1},m_{2}} \right)}}},$

respectively. To quantify the benefit of cooperation in the case wherethe CF input links are rate-limited, a more general notion ofprobability of error is required, which is described next.

For i∈{1,2}, fix r_(i)≧0, and set

K _(i)=min{└e ^(nr) ^(i) ┘,M _(i)}

L=└M _(i) /K _(i)┘.

Furthermore, for i∈{1,2} and k_(i)∈[K_(i)], define the set S_(i,k) _(i)⊂[M_(i)] of size L_(i) as

S _(i,k) _(i) ={k _(i)−1)L _(i)+1, . . . ,k _(i) L _(i)}.  (18)

To simplify notation, denote S_(i,k) _(i) with S_(k) _(i) . Now define

${{P_{e}^{(n)}\left( {r_{1},r_{2}} \right)} = {\min\limits_{\sigma_{1},\sigma_{2}}{\max\limits_{k_{1},k_{2\;}}{\frac{1}{L_{1}L_{2}}{\sum\limits_{\underset{m_{2} \in S_{k_{2}}}{m_{1} \in S_{k_{1}}}}{\lambda_{n}\left( {{\sigma_{1}\left( m_{1} \right)},{\sigma_{2}\left( m_{2} \right)}} \right)}}}}}},$

where the minimum is over all permutations σ₁ and σ₂ of the sets [M₁]and [M₂], respectively. To compute P_(e) ^((n))(r₁,r₂), partition thematrix

Λ_(n):=(λ_(n),(m ₁ ,m ₂))_(m) ₁ _(m) ₂   (19)

into K₁K₂ blocks of size L₁×L₂. Then calculate the average of theentries within each block. The (r₁, r₂)-probability of error, P_(e)^((n))(r₁,r₂), is the maximum of the K₁K₂ obtained average values. Theminimization over all permutations of the rows and columns of Λ_(n)ensures P_(e) ^((n))(r₁,r₂) is invariant with respect to relabeling themessages. Note that P_(e,avg) ^((n)) and P_(e,max) ^((n)) are specialcases of P_(e) ^((n))(r₁,r₂), since

P _(e) ^((n))(0,0)=P _(e,avg) ^((n))

and for sufficiently large values of r₁ and r₂,

P _(e) ^((n))(r ₁ ,r ₂)=P _(e,max) ^((n)).

A rate pair (R₁,R₂) is achievable under the (r₁,r₂) notion of error fora MAC with a (C_(in),C_(out))−CF and J steps of cooperation if for all∈,δ>0, and for n sufficiently large, there exists an (n,M₁,M₂,J)-codesuch that for i∈{1,2}.

$\begin{matrix}{{{\frac{1}{n}{\log \left( {K_{i}L_{i}} \right)}} \geq {R_{i} - \delta}},} & (20)\end{matrix}$

and P_(e) ^((n))(r₁,r₂)≦∈. In Equation (20), use K_(i)L_(i) instead ofM_(i) since only K_(i)L_(i) elements of [M_(i)] are used in calculatingP_(e) ^((n))(r₁,r₂). Define the (r₁,r₂)-capacity region as the closureof the set of all rates that are achievable under the (r₁,r₂) notion oferror.

3.2 Cooperation and Reliability

Define the nonnegative numbers R*₁ and R*₂ as the maximum of R₁ and R₂over the average-error capacity region of a MAC with a(C_(in),C_(out))−CF and J cooperation steps. From the capacity region ofthe MAC with conferencing encoders, it follows

$R_{1}^{*} = {\max\limits_{X_{1} - U - X_{2}}{\min \left\{ {{{I\left( {{X_{1};\left. Y \middle| U \right.},X_{2}} \right)} + C_{12}},{I\left( {X_{1},{X_{2};Y}} \right)}} \right\}}}$${R_{2}^{*} = {\max\limits_{X_{1} - U - X_{2}}{\min \left\{ {{{I\left( {{X_{2};\left. Y \middle| U \right.},X_{1}} \right)} + C_{21}},{I\left( {X_{1},{X_{2};Y}} \right)}} \right\}}}},$

where C₁₂=min{C_(in) ¹,C_(out) ²} and C₂₁=min{C_(in) ²,C_(out) ¹}. Thisfollows from the fact that when one encoder transmits at rate zero,cooperation through a CF is no more powerful then direct conferencing.Note that R*₁ and R*₂ do not depend on J, since using multipleconferencing steps does not enlarge the average-error capacity regionfor the 2-user MAC.

The main result of this section is stated next, which says that if arate pair is achievable for a MAC with a CF under average error, thensufficiently increasing the capacities of the CF links ensures that thesame rate pair is also achievable under a stricter notion of error. Thisresult applies to any memoryless MAC whose average-error capacity regionis bounded.

Theorem 9.

The ({tilde over (r)}₁,{tilde over (r)}₂)-capacity region of a MAC witha ({tilde over (C)}_(in),{tilde over (C)}_(out))−CF and {tilde over (J)}steps of cooperation contains the average-error capacity region of thesame MAC with a (C_(in),C_(out))−CF and J steps of cooperation if {tildeover (J)}≧J+1 and for i∈{1, 2},

{tilde over (C)} _(in)>min{C _(in) ^(i) +{tilde over (r)} _(i) ,R* _(i)}

{tilde over (C)} _(out) >C _(out) ^(i).

Furthermore, if for i∈{1,2},{tilde over (C)}_(in) ^(i)>R*_(i), itsuffices to take {tilde over (J)}=1. Similarly, {tilde over (J)}=1 issufficient when C_(in)=0.

A detailed proof can be found in Parham Noorzad. Michelle Effros, andMichael Langberg, Can Negligible Cooperation Increase NetworkReliability? (Extended Version), arxiv.org/abs/1601.05769, 21 Jan. 2016,which is hereby incorporated by reference in its entirety.

3.3 The Average- and Maximal-Error Capacity Regions

For every (C_(in),C_(out))∈

, let

(C_(in),C_(out)) denote the average-error capacity region of a MAC witha (C_(in),C_(out))−CF with J cooperation steps. Let

(C_(in),C_(out)) denote the convex closure of

$\underset{J = 1}{\bigcup\limits^{\infty}}{{\left( {C_{i\; n},C_{out}} \right).}}$

Define

and

similarly.

Next, a generalization of the notion of sum-capacity is introduced whichis useful for the results of this section. Let

be a compact subset of

. For every α∈[0,1] define

$\begin{matrix}{{C^{\alpha}{()}} = {\max\limits_{{({x,y})} \in}{\left( {{\alpha \; x} + {\left( {1 - \alpha} \right)y}} \right).}}} & (21)\end{matrix}$

Note that C^(α) is the value of the support function of

computed with respect to the vector (α,1−α). When

is the capacity region of a network. C^(1/2)(

) equals half the corresponding sum-capacity.

For every C_(out)∈

, let

(C _(out))=

((∞,∞),C _(out)).

and for α∈[0,1], define

C _(avg) ^(α)(C _(out))=C ^(α)(

_(avg)(C _(out))).

In words,

(C_(out)) denotes the average-error capacity region of a MAC with a CFthat has access to both messages and output edge capacities given byC_(out)=(C_(out) ¹,C_(out) ²). Define

(C_(out)) and C_(max) ^(α)(C_(out)) similarly. Note that in thedefinitions of both

(C_(out)) and

(C_(out)), C_(in)=(∞,∞) can be replaced with any C_(in)=(C_(in) ¹,C_(in)²) where

${\min \left\{ {C_{i\; n}^{1},C_{i\; n}^{2}} \right\}} > {\max\limits_{p{({x_{1},x_{2}})}}{{I\left( {X_{1},{X_{2};Y}} \right)}.}}$

The “avg” and “max” subscripts are dropped when a statement is true forboth the maximal- and average-error capacity regions.

The main theorem of this section follows. This theorem states thatcooperation through a CF that has access to both messages results in anetwork whose maximal- and average-error capacity regions are identical.

Theorem 10.

For every C_(out)∈

,

(C _(out))=

(C _(out)).

Furthermore, for Dueck's contraction MAC, there exists C_(in)∈

such that for every C_(out)∈

(C_(in),C_(out)) is a proper subset of

(C_(in),C_(out)).

Next, the capacity region of a network containing edges of negligiblecapacity is formally defined. Let

be a network containing a single edge of negligible capacity. For everyδ>0, let

be the same network with the difference that the edge with negligiblecapacity is replaced

with an edge of capacity δ. A rate vector is achievable over

if and only if for every δ>0, that rate vector is achievable over

(δ). Achievability over networks with multiple edges of negligiblecapacity is defined inductively.

From the above definition, it now follows that the capacity region of aMAC with a CF that has complete access to both messages and output edgesof negligible capacity, equals

$\bigcap\limits_{C_{out} \in {\mathbb{R}}_{> 0}^{2}}{{\left( C_{out} \right).}}$

From Theorem 10 it follows that for every MAC.

⋂ C out ∈ ℝ > 0 2  ma   x  ( C out ) = ⋂ C out ∈ ℝ > 0 2  avg  ( Cout ) ⊇ avg  ( 0 ) , ( 22 )

where 0=(0,0). Thus if for a MAC we have

C _(avg) ^(α)(0)>C _(max) ^(α)(0)  (23)

for some α∈(0,1), then C_(max) ^(α)(C_(out)) is not continuous atC_(out)=0, since by Equation (22),

${\lim\limits_{C_{out}->0^{+}}{C_{{ma}\; x}^{\alpha}\left( \left( {C_{out},C_{out}} \right) \right)}} = {{\lim\limits_{C_{out}->0^{+}}{C_{avg}^{\alpha}\left( \left( {C_{out},C_{out}} \right) \right)}} \geq {C_{avg}^{\alpha}(0)} > {{C_{{ma}\; x}^{\alpha}(0)}.}}$

It can be shown that Dueck's contraction MAC satisfies Equation (23) forevery α∈(0,1). Thus there exists a MAC where C_(max) ^(α)(C_(out)) isnot continuous at C_(out)=0 for any α∈(0,1). This example demonstratesthat the introduction of a negligible capacity edge can have a strictlypositive impact on the network capacity.

For the average-error capacity region of the MAC, less is known. Forsome MACs, the directional derivative of C_(avg) ^(1/2)(C_(out)) atC_(out)=0 equals infinity for all unit vectors in

. The question of whether C_(avg) ^(1/2)(C_(out)) is continuous on

for such MACs remains open.

Next, an overview of the proof of the first part of Theorem 10 isprovided. First, using Theorem 9, we show that for everyC_(out)=(C_(out) ¹,C_(out) ²) and {tilde over (C)}_(out)=({tilde over(C)}_(out) ¹,{tilde over (C)}_(out) ²) in

for which {tilde over (C)}_(out) ¹>C_(out) ¹ and {tilde over (C)}_(out)²>{tilde over (C)}_(out) ², we have

({tilde over (C)} _(out))⊃

(C _(out)).

Note that

(C_(out)) contains

(C_(out)). Thus a continuity argument may be helpful in proving equalitybetween the average- and maximal-error capacity regions. Since studyingC^(α) is simpler than studying the capacity region directly, weformulate our problem in terms of C^(α). For every α∈[0,1], we have

C _(max) ^(α)(C _(out))≦C _(avg) ^(α)(C _(out))≦C _(max) ^(α)({tildeover (C)} _(out)).  (24)

The next theorem investigates the continuity of the C^(α)'s.

Theorem 11.

For every α∈[0,1], the mappings C_(max) ^(α)(C_(out)) and C_(avg)^(α)(C_(out)) are concave on

and thus continuous on

.

By combining the above theorem with Equation (24), it follows that forevery α∈[0,1] and C_(out)∈

.

C _(max) ^(α)(C _(out))=C _(avg) ^(α)(C _(out)).

Since for a given capacity region

, the mapping α

C_(α)(

) characterizes

precisely (see next theorem), for every C_(out)

,

(C _(out))=

(C _(out)).

Theorem 12.

Let

⊂

be non-empty, compact, convex, and closed under projections onto theaxes, that is, if (x,y) is in

, then so are (x,0) and (0,y). Then

={(x,y)∈

|∀α∈[0,1]:αx+(1−α)y≦C ^(α)}.

This result continues to hold for subsets

for any positive integer k.

Next the maximal- and average-error capacity regions of the MAC withconferencing are studied. Let (C₁₂,C₂₁)∈

and

(C₁₂,C₂₁) denote the maximal- or average-error capacity region of theMAC with (C₁₂,C₂₁)-conferencing. Then for every (C₁₂,C₂₁)∈

,

(C ₁₂ ,C ₂₁)=

(C _(in) ,C _(out)),  (25)

where

C _(in)=(C ₁₂ ,C ₂₁)  (26)

C _(out)=(C ₂₁ ,C ₁₂)  (27)

Equation (25) follows from the fact that for a CF whose input and outputlink capacities are given by Equations (26) and (27), the strategy wherethe CF forwards its received information from one encoder to the otheris optimal.

For every α∈[0,1], define

C _(conf) ^(α)(C ₁₂ ,C ₂₁)=C ^(α)(C _(in) ,C _(out)),  (28)

where C_(in) and C_(out) are given by Equations (26) and (27). The nextresult considers the continuity of C_(conf) ^(α) for various values ofα∈[0,1].

Theorem 13.

For every α∈[0,1], C_(conf,avg) ^(α) is continuous on

and C_(conf,max) ^(α) is continuous on

. In addition, C_(conf,max) ^(1/2) is continuous at the point (0,0) aswell.

Finally, note that the second part of Theorem 10 implies that thereexists a MAC where for some (C₁₂,C₂₁)∈

,

_(,max)(C₁₂,C₂₁)) is a proper subset of

_(,avg)(C₁₂,C₂₁). Thus direct cooperation via conferencing does notnecessarily lead to identical maximal- and average-error capacityregions.

Although the present invention has been described in certain specificaspects, many additional modifications and variations would be apparentto those skilled in the art. For example, the discussion proivded abovereferences use of cooperation facilitators in the context of GuassianMACs. Cooperation facilitators in accordance with various embodiments ofthe invention can improve network performance in a variety of contextsinvolving shared resources. Furthermore, although specific techniquesfor building code books are described above, the processes presentedherein can be utilized to generate code books that can be readilyimplemented in encoders used in typical communication devices to achievelow latency encoding of message data based upon data received fromcommunication facilitators. It is therefore to be understood that thepresent invention can be practiced otherwise than specifically describedwithout departing from the scope and spirit of the present invention.Thus, embodiments of the present invention should be considered in allrespects as illustrative and not restrictive. Accordingly, the scope ofthe invention should be determined not by the embodiments illustrated,but by the appended claims and their equivalents.

What is claimed is:
 1. A communication system, comprising: a pluralityof nodes, that each comprise: a transmitter; a receiver; and an encoderthat encodes message data for transmission using a plurality ofcodewords: a cooperation facilitator node comprising: a transmitter; anda receiver; wherein the plurality of nodes are configured to transmitdata parameters to the cooperation facilitator; wherein the cooperationfacilitator is configured to generate cooperation parameters based uponthe data parameters received from the plurality of nodes; wherein thecooperation facilitator is configured to transmit cooperation parametersto the plurality of nodes; and wherein the encoder in each of theplurality of nodes selects a codeword from the plurality of codewordsbased at least in part upon the cooperation parameters received from thecommunication facilitator and transmit the selected codeword via themultiple access channel.
 2. The communication system of claim 1, whereina sum-capacity of the communication system achieved using codewordsselected at least in part based upon the cooperation parameters receivedfrom the cooperation facilitator is greater than the sum-capacity of thecommunication system achieved when each of the plurality of encodersencodes data without communicating with a cooperation facilitator. 3.The communication system of claim 1, wherein a reliability of thecommunication system achieved using codewords selected at least in partbased upon the cooperation parameters received from the cooperationfacilitator is greater than the reliability of the communication systemachieved with each of the plurality of encoders encodes data withoutcommunicating with a cooperation facilitator.
 4. The communicationsystem of claim 1, wherein the cooperation parameters includeconferencing parameters.
 5. The communication system of claim 1, whereinthe cooperation parameters include coordinating parameters.
 6. Thecommunication system of claim 1, wherein the transmitter in each of theplurality of nodes transmits data via a multiple access channel.
 7. Thecommunication system of claim 6, wherein the multiple access channel isa shared wireless channel.
 8. The communication system of claim 6,wherein the multiple access channel is a Gaussian multiple accesschannel.
 9. The communication system of claim 1, wherein the pluralityof nodes is two nodes.
 10. The communication system of claim 1, whereinthe plurality of nodes is at least three nodes.
 11. The communicationsystem of claim 1, wherein the transmitter in each of the plurality ofnodes transmits to a plurality of receivers.
 12. The communicationsystem of claim 1, wherein the cooperation facilitator generatesmultiple rounds of cooperation parameters prior to codewordtransmission.
 13. The communication system of claim 1, whereincooperation parameters are transmitted to the plurality of nodes by thecoordination facilitator via a separate channel to a channel on whichone or more of the plurality of nodes transmit codewoards selected atleast in part based upon the cooperation parameters received from thecooperation facilitator.
 14. A cooperation facilitator, comprising: atransmitter; a receiver; and a cooperation facilitator controller;wherein the cooperation facilitator controller is configured to receivedata parameters from a plurality of nodes; wherein the cooperationfacilitator is configured to generate cooperation parameters based uponthe data parameters received from the plurality of nodes; and whereinthe cooperation facilitator is configured to transmit cooperationparameters to the plurality of nodes that enable encoders in each of theplurality of nodes to select a codeword from a plurality of codewordsfor transmission.
 15. The communication system of claim 14, wherein asum-capacity of a portion of a communication network including thecooperation facilitator achieved by encoders in each of the plurality ofnodes using codewords selected at least in part based upon thecooperation parameters received from the cooperation facilitator isgreater than the sum-capacity of the portion of a communication networkachieved when each of the plurality of encoders encodes data withoutcommunicating with a cooperation facilitator.
 16. The communicationsystem of claim 14, wherein a reliability of a portion of acommunication network including the cooperation facilitator achieved byencoders in each of the plurality of nodes using codewords selected atleast in part based upon the cooperation parameters received from thecooperation facilitator is greater than the reliability of the portionof a communication network achieved with each of the plurality ofencoders encodes data without communicating with a cooperationfacilitator.
 17. The communication system of claim 14, wherein thecooperation parameters include conferencing parameters.
 18. Thecommunication system of claim 14, wherein the cooperation parametersinclude coordinating parameters.
 19. The communication system of claim14, wherein a transmitter in each of the plurality of nodes transmitsdata via a multiple access channel.
 20. The communication system ofclaim 17, wherein the multiple access channel is a shared wirelesschannel.
 21. The communication system of claim 17, wherein the multipleaccess channel is a Gaussian multiple access channel.
 22. Thecommunication system of claim 14, wherein the plurality of nodes is twonodes.
 23. The communication system of claim 14, wherein the pluralityof nodes is at least three nodes.
 24. The communication system of claim14, wherein the cooperation facilitator generates multiple rounds ofcooperation parameters prior to codeword transmission.